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THE 



American Science Series 

FOR SCHOOLS AND COLLEGES. 



The principal objects of the series are to supply the lack— in some subjects 
very great— of authoritative books whose principles are, so far as practicable, 
illustrated by familiar American facts, and also to supply the other lack that 
the advance of Science perennially creates, of text-books which at least do not 
contradict the latest generalizations. The books of this series systematically 
outline the field of Science, as the term is usually employed with reference to 
general education. The scheme includes an Advanced Course, a Briefer Course, 
and an Elementary Course. 

^p° In ordering be careful to state which course is desired— Advanced, 
Briefer, or Elementary. 



Physics, 

By George F. Barker, Professor 
in the University of Pennsylvania. 
In Preparation. 

Chemistry. 

By Ira Remsen, Professor in the 
Johns Hopkins University. 
Briefer Course, 387 pp. 
Elementary Course, 272 pp. 

Astronomy. 

By Simon Newcomb, Professor in 
the Johns Hopkins University, and 
Edward S. Holden, Director of the 
Lick Observatory. 

Advanced Course, 512 pp. 

Briefer Course, 352 pp. 

Biology. 

By William T. Sedgwick, Pro- 
fessor in the Massachusetts Insti- 
tute of Technology, and Edmund 
B. Wilson, Professor in Bryn Mawr 
College. 

Part I. — Introductory, 193 pp. 

Botany. 

By C. E. Besset, Professor in the 
University of Nebraska ; formerly 
in the Iowa Agricultural College* 

Advanced Course, 611 jap. 

Briefer Course, 292 pp. 



Zoology. 

By A. S. Packard, Professor 
of Zoology and Geology in Brown 
University. 

Advanced Course, 722 pp. 
Bi-iefer Course, 338 pp. 
Elementary Course, 290 pp. 

The Human Body. 

By H. Newell Martin, Profes- 
sor in the Johns Hopkins Univer- 
sity. 

Advanced Course, 621 + 34 pp. 
Copies without the Appendix on 
Reproduction will be sent when 
specially ordered. 

Briefer Course, 377 pp. 

Elemental Course, 261 pp. 



Political Economy. 

By Francis A. Walker 
dent Massachusetts 
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Advanced Course, 490 pp 

Bi-iefer Course, 415 pp. 



Institute of 



HENRY HOLT & CO., Publishers, NEW YORK. 




THE PLANET JUPITER. 
As seen with the 26-inch telescope at Washington, 1875. June 



AMEEICAR SCIENCE SERIES— ADVANCED COURSE 



ASTRONOMY 



FOR 



HIGH SCHOOLS AND COLLEGES 



BY 

SIMON NEWCOMB, LL.D., 

PROFESSOR OP MATHEMATICS IN THE JOHNS HOPKINS UNIVERSITY, 
AND 

EDWARD S. HOLDER LL.D., 

DIRECTOR OP THE LICK OBSERVATORY. 



SIXTH EDITION REVISED 




NEW YORK 
HENKY HOLT AND COMPANY 

18.89 




Q&13 



■A 



Copyright, 1879, 

BY 

Henry Holt & Co. 



PREFACE 



The following work is designed principally for the use 
of those who desire to pursue the study of Astronomy as a 
branch of liberal education. To facilitate its use by stu- 
dents of different grades, the subject-matter is divided into 
two classes, distinguished by the size of the type. The 
portions in large type form a complete course for the use 
of those who desire only such a general knowledge of the 
subject as can be acquired without the application of ad- 
vanced mathematics. Sometimes, especially in the ear- 
lier chapters, a knowledge of elementary trigonometry 
and natural philosophy will be found necessary to the full 
understanding of this course, but it is believed that it can 
nearly all be mastered by one having at command only 
those geometrical ideas which are familiar to most intelli- 
gent students in our advanced schools. 

The portions in small type comprise additions for the 
use of those students who either desire a more detailed 
and precise knowledge of the subject, or who intend to 
make astronomy a special study. In this, as in the ele- 
mentary course, the rule has been never to use more ad- 
vanced mathematical methods than are necessary to the 
development of the subject, but in some eases a knowl- 
edge of Analytic Geometry, in others of the Differential 
Calculus, and in others of elementary Mechanics, is neces- 



vi PREFACE. 

sarily presupposed. The object aimed at has been to lay 
a broad foundation for further study rather than to at- 
tempt the detailed presentation of any special branch. 

As some students, especially in seminaries, may not de- 
sire so extended a knowledge of the subject as that em- 
braced in the course in large type, the following hints are 
added for their benefit : Chapter I., on the relation of the 
earth to the heavens, Chapter III., on the motion of the 
earth, and the chapter on Chronology should, so far as pos- 
sible, be mastered by all. The remaining parts of the course 
may be left to the selection of the teacher or student. 
Most persons will desire to know something of the tele- 
scope (Chapter II.), of the arrangement of the solar system 
(Chapter IV. , §§ 1-2, and Part II. , Chapter II.), of eclipses, 
of the phases of the moon, of the physical constitution of 
the sun (Part II., Chapter II.), and of the constellations 
(Part III., Chapter I.). It is to be expected that all will 
be interested in the subjects of the planets, comets, and 
meteors, treated in Part II. , the study of which involves 
no difficulty. 

An acknowledgment is due to the managers of the 
Clarendon Press, Oxford, who have allowed the use of a 
number of electrotypes from Chambers's Descriptive 
Astronomy. Messrs. Fauth & Co. , instrument-makers, of 
Washington, have also lent electrotypes of instruments, 
and a few electrotypes have been kindly furnished by the 
editors of the American Journal of Science and of the 
Popular Science Monthly, The greater part of the illus- 
trations have, however, been prepared expressly for the 
work. 



CONTENTS 



PART I 



PAGB 

Introduction 1 



CHAPTER I. 

THE RELATION OF THE EARTH TO THE HEAVENS. 

The Eartli — The Celestial Sphere — Relation of the Sphere to the 
Horizon — Plane of the Horizon — The Diurnal Motion — The 
Diurnal Motion in different Latitudes — Correspondence of the 
Terrestrial and Celestial Spheres — Right Ascension and Dec- 
lination — Relation of Time to the Sphere — Determination of 
Terrestrial Longitudes — Mathematical Theory of the Celestial 
Sphere — Determination of Latitudes on the Eartli by Astro- 
nomical Observations — Parallax and Semidiameter 9 

CHAPTER II. 

ASTRONOMICAL INSTRUMENTS. 

The Refracting Telescope — Reflecting Telescopes — Chronometers 
and Clocks — The Transit Instrument — Graduated Circles — 
The Meridian Circle — The Equatorial — The Zenith Telescope 
—The Sextant 53 

CHAPTER HI. 

MOTION OF THE EARTn. 

Ancient Ideas of the Planets — Annual Revolution of the Earth — 
The Sun's apparent Path— Obliquity of the Ecliptic — The 
Seasons 96 

CHAPTER IV. 

THE PLANETARY MOTIONS. 

Apparent and Real Motions of the Planets— Gravitation in the 

Heavens — Kepler's Laws of Planetary Motion 113 



vin CONTENTS. 

CHAPTER V. 

UNIVERSAL GRAVITATION. 

PAGE 

Newton's Laws of Motion — Problems of Gravitation — Results of 

Gravitation — Remarks on tlie Theory of Gravitation 131 

CHAPTER VI. 

THE MOTION AND ATTRACTION OP THE MOON. 

The Moon's Motion and Phases — The Sun's disturbing Force- 
Motion of the Moon's Nodes — Motion of the Perigee — Rotation 
of the Moon— The Tides 152 

CHAPTER VII. 

ECLIPSES OF THE SUN AND MOON. 

The Earth's Shadow and Penumbra — Eclipses of the Moon- 
Eclipses of the Sun— The Recurrence of Eclipses — Character 
of Eclipses 168 

CHAPTER VIII. 

THE EARTH. 

Mass and Density of the Earth — Laws of Terrestrial Gravitation — 
Figure and Magnitude of the Earth — Change of Gravity with 
the Latitude — Motion of the Earth's Axis, or Precession of the 
Equinoxes 188 

CHAPTER IX. 

CELESTIAL MEASUREMENTS OF MASS AND DISTANCE. 

The Celestial Scale of Measurement — Measures of the Solar 

Parallax — Relative Masses of the Sun and Planets 213 

CHAPTER X. 

THE REFRACTION AND ABERRATION OF LIGHT. 

Atmospheric Refraction — Aberration and the Motion of Light 234 

CHAPTER XI. 
CHRONOLOGY. 

Astronomical Measures of Time — Formation of Calendars — 
Division of the Day — Remarks on improving the Calendar — 
The Astronomical Ephemeris or Nautical Almanac 245 



CONTENTS. ix 

PAET II. 

THE SOLAR SYSTEM IN DETAIL. 



CHAPTER I. 

PAGE 

Structure op the Solar System 267 



CHAPTER II. 

THE SUN. 

General Summary — The Photosphere — Sun-Spots and Faculse — 
The Sun's Chromosphere and Corona — Sources of the Sun's 
Heat 278 

CHAPTER III. 

THE INFERIOR PLANETS. 

Motions and Aspects — Aspect and Rotation of Mercury — The 
Aspect and supposed Rotation of Venus — Transits of Mercury 
and Venus — Supposed intramercurial Planets 310 

CHAPTER IV. 
The Moon 326 

CHAPTER V. 

THE PLANET MARS. 
The Description of the Planet — Satellites of Mars 834 

CHAPTER VI. 
The Minor Planets 340 

CHAPTER VII. 

JUPITER AND HIS SATELLITES. 

The Planet Jupiter— The Satellites of Jupiter 343 

CHAPTER VIII. 

SATURN AND HIS SYSTEM. 

General Description — The Rings of Saturn— Satellites of Saturn. . 353 



X CONTENTS. 

CHAPTER IX. 

PAGE 

The Planet Uranus— Satellites of Uranus 362 

CHAPTER X. 
The Planet Neptune— Satellite of Neptune 365 

CHAPTER XI. 
The Physical Constitution of the Planets 370 

CHAPTER XII. 

METEORS. 

Phenomena and Causes of Meteors — Meteoric Showers 375 

CHAPTER XIII. 

COMETS. 

Aspect of Comets — The Vaporous Envelopes — The Physical Con- 
stitution of Comets — Motion of Comets — Origin of Comets — 
Remarkable Comets 388 



PART III. 

THE UNIVERSE AT LARGE. 



Introduction 411 

CHAPTER I. 

THE CONSTELLATIONS. 

General Aspect of the Heavens — Magnitude of the Stars — The 
Constellations and Names of the Stars — Description of Con- 
stellations — Numbering and Cataloguing the Stars 415 

CHAPTER II. 

VARIABLE AND TEMPORARY STARS. 

Stars Regularly Variable— Temporary or New Stars— Theory of 
Variable Stars -' -440 



CONTENTS. xi 

CHAPTER III. 

MULTIPLE STARS. 

PAGB 

Character of Double and Multiple Stars — Orbits of Binary Stars. . 448 
CHAPTER IV. 

NEBULAE AND CLUSTERS. 

Discovery of Nebulae — Classification of Nebulae and Clusters — 
Star Clusters — Spectra of Nebulae and Clusters — Distribution 
of Nebulae and Clusters on the Surface of the Celestial 
Sphere 457 

CHAPTER V. 

SPECTRA OP FIXED STARS. 

Characters of Stellar Spectra — Motion of Stars in the Line of Sight. 468 
CHAPTER VI. 

MOTIONS AND DISTANCES OF THE STARS. 

Proper Motions — Proper Motion of the Sun — Distances of the 

Fixed Stars 472 

CHAPTER VII. 
Construction of the Heavens 478 

CHAPTER VIII. 
Cosmogony 492 

Index 503 



ASTRONOMY 



INTRODUCTION. 

Astronomy (atftrfp — a star, and radios — a law) is the 
science which has to do with the heavenly bodies, their 
appearances, their nature, and the laws governing their 
real and their apparent motions. 

In approaching the study of this, the most ancient of the 
sciences depending upon observation, it must be borne in 
mind that its progress is most intimately connected with 
that of the race, it having always been the basis of geog- 
raphy and navigation, and the soul of chronology. Some 
of the chief advances and discoveries in abstract mathe- 
matics have been made in its service, and the methods 
both of observation and analysis once peculiar to its prac- 
tice now furnish the firm bases upon which rest that great 
group of exact sciences which we call physics. 

It is more important to the student that he should be- 
come penetrated with the spirit of the methods of astron- 
omy than that he should recollect its minutiae, and it is 
most important that the knowledge which he may gain 
from this or other books should be referred by him to its 
true sources. For example, it will often be necessary to 
speak of certain planes or circles, the ecliptic, the equa- 
tor, the meridian, etc., and of the relation of the appa- 
rent positions of stars and planets to them ; but his labor 
will be useless if it has not succeeded in giving him a 
precise notion of these circles and planes as they exist in 



% ASTRONOMY. 

the sky, and not merely in the figures of his text -book. 
Above all, the study of this science, in which not a single 
step could have been taken without careful and painstak- 
ing observation of the heavens, should lead its student 
himself to attentively regard the phenomena daily ' and 
hourly presented to him by the heavens. 

Does the sun set daily in the same point of the hori- 
zon ? Does a change of his own station affect this and 
other aspects of the sky ? At what time does the full 
moon rise ? Which way are the horns of the young 
moon pointed ? These and a thousand other questions 
are already answered by the observant eyes of the an- 
cients, who discovered not only the existence, but the 
motions, of the various planets, and gave special names to 
no less than fourscore stars. The modern pupil is more 
richly equipped for observation than the ancient philoso- 
pher. If one could have put a mere opera-glass in the 
hands of Hipparchus the world need not have waited two 
thousand years to know the nature of that early mystery, 
the Milky Way, nor would it have required a Galileo to 
discover the phases of Venus and the spots on the sun. 

From the earliest times the science has steadily progress- 
ed by means of faithful observation and sound reasoning 
upon the data which observation gives. The advances in 
our special knowledge of this science have made it con- 
venient to regard it as divided into certain portions, which 
it is often convenient to consider separately, although the 
boundaries cannot be precisely fixed. 

Spherical and Practical Astronomy — First in logical 
order we have the instruments and methods by which the 
positions of the heavenly bodies are determined from obser- 
vation, and by which geographical positions are also fixed. 
The branch which treats of these is called spherical and 
practical astronomy. Spherical astronomy provides the 
mathematical theory, and practical astronomy (which is 
almost as much an art as a science) treats of the applica- 
tion of this theory. 



DIVISIONS OF THE SUBJECT. 3 

Theoretical Astronomy deals with the laws of motion of 
the celestial bodies as determined by repeated observations 
of their positions, and by the laws according to which they 
ought to move under the influence of their mutual gravi- 
tation. The purely mathematical part of the science, by 
which the laws of the celestial motions are deduced from 
the theory of gravitation alone, is also called Celestial 
Mechanics, a term first applied by La Place in the title of 
his great work Mecanique Celeste. 

Cosmical Physics.— A third branch which has received 
its greatest developments in quite recent times may be 
called Cosmical Physics. Physical astronomy might be 
a better appellation, were it not sometimes applied to 
celestial mechanics. This branch treats of the physical 
constitution and aspects of the heavenly bodies as investi- 
gated with the telescope, the spectroscope, etc. 

We thus have three great branches which run into each 
other by insensible gradations, but undei which a large 
part of the astronomical research of the present day may 
be included. In a work like the present, however, it 
will not be advisable to follow strictly this order of sub- 
jects ; we shall rather strive to present the whole subject 
in the order in which it can best be understood. This 
order will be somewhat like that in which the knowl- 
edge has been actually acquired by the astronomers of 
different ages. 

Owing to the frequency with which we have to use 
terms expressing angular measure, or referring to circles 
on a sphere, it may be admissible, at the outset, to give 
an idea of these terms, and to recapitulate some prop- 
erties of the sphere. 

Angular Measures. — The unit of angular measure most 
used for considerable angles, is the degree, 360 of which 
extend round the circle. The reader knows that it is 90° 
from tiie horizon to the zenith, and that two objects 180° 
apart are diametrically opposite. An idea of distances of 



4 ASTRONOMY. 

a few degrees may be obtained by looking at the two stars 
which form the pointers in the constellation Ursa Major 
(the Dipper), soon to be described. These stars are 5° 
apart. The angular diameters of the sun and moon are 
each a little more than half a degree, or 30'. 

An object subtending an angle of only one minute ap- 
pears as a point rather than a disk, but is still plainly vis- 
ible to the ordinary eye. Helmholtz finds that if two 
minute points are nearer together than about V 12", no 
eye can any longer distinguish them as two. If the ob- 
jects are not plainly visible — if they are small stars, for 
instance, they may have to be separated 3', 5', or even 
10', to be seen as separate objects. Kear the star a LyrcB 
are a pair of stars Z\' apart, which can be separated only 
by very good eyes. 

If the object be not a point, but a long line, it may be 
Been by a good eye when its breadth subtends an angle of 
only a fraction of a minute ; the limit probably ranges 
from 10" to 15". 

If the object be much brighter than the background on 
which it is seen, there is no limit below which it is neces- 
sarily invisible. Its visibility then depends solely on the 
quantity of light which it sends to the eye. It is not 
likely that the brightest stars subtend an angle of yj-^ of 
a second. 

So long as the angle subtended by an object is small, we 
may regard it as varying directly as the linear magnitude 
of the body, and inversely as its distance from the ob- 
server. A line seen perpendicularly subtends an angle 
of 1° when it is a little less than 60 times its length dis- 
tant from the observer (more exactly when it is 57-3 
lengths distant) ; an angle of 1' when it is 3438 lengths 
distant, and of V when it is 206265 lengths distant. 
These numbers are obtained by dividing the number of 
degrees, minutes, and seconds, respectively, in the cir- 
cumference, by 2 x 3-11159265, the ratio of the circum- 
' ference of a circle to the radius. 



PLANES AND CIRCLES OF A SPHERE. 5 

Planes and Circles of a Sphere. — Let Fig. 1 represent 
the outline of a sphere, of which is the centre. Imagine 
a plane A B to pass through the centre and cut the 
sphere. This plane will divide the sphere into two equal 
parts called hemispheres. It will intersect the sphere in 
a circle A E B F, called a great circle of the sphere. 




Fig. 1.— sections of a sphere by planes. 

Through let a straight line P P' be passed per- 
pendicular to the plane. The points P and P\ in which 
it intersects the surface of the sphere, are everywhere 90° 
from the circle A E B F. They are called poles of that 
circle. 

Imagine another plane C E D F, to cut the sphere in a 
great circle. Its poles will be Q and Q'. 

The following relations between the angles made by the 
figures will then hold : 

I. The angle P Q between the poles will be equal to the 
inclination of the planes to each other. 

II. The arc B D, which measures the greatest distance 
between the two great circles, will be equal to this same 
inclination. 

III. The points E and F, in which the two great circles 
intersect each oilier , are the poles of the great circle P Q A 
OP' Q'B I), which pass th rough the poles of 'the first circle. 



SYMBOLS AND ABBREVIATIONS. 



SIGNS OP THE PLANETS, ETC. 



The Sun. 
The Moon. 
Mercury. 
Venus. 



or $ The Earth. 



$ Mars. 

11 Jupiter. 

^ Saturn. 

£) Uranus, 

tj? Neptune. 



The asteroids are distinguished by a circle inclosing a number, which 
number indicates the order of discovery, or by their names, or by both, 
as (100); Hecate 



SIGNS OP THE ZODIAC. 



Spring 
signs. 

Summer 
signs. 



1. 


V Aries. 


2. 


b Taurus. 


3. 


n Gemini. 


4. 


g£ Cancer. 


5. 


SI Leo. 


6. 


TTH. Virgo. 



Autumn 
signs. 



7. =g= Libra. 

8. tii Scorpius. 

9. t Sagittarius. 
Winter { J* * Capricorn** 

sisrns i U - ™ Muarius. 

SlgDS * ( 12. x Pisces. 



ASPECTS. 



6 Conjunction, or having the same longitude or right ascension. 
a Quadrature, or differing 90° in " " " 

8 Opposition, or differing 180° in " '* " 



ASTRONOMICAL SYMBOLS. 

MISCELLANEOUS SYMBOLS. 



Q Ascending node. 
t5 Descending node. 
N. North. S. South. 
E. East. W. West. 

° Degrees. 

' Minutes of arc. 

" Seconds of arc. 

h Hours. 

m Minutes of time. 

8 Seconds of time. 
L, Mean longitude of a body. 
g, Mean anomaly. 
f, True anomaly. 
n, Mean sidereal motion in a 

of time. 
7% Radius vector. 
0, Angle of eccentricity. 
it, Longitude of perihelion 
parallax). 

p, Earth 



MA or a, Right ascension. 
Dec. or 6, Declination. 
C, True zenith distance. 
{', Apparent zenith distance. 
A Distance from the earth. 
I, Heliocen'ric longitude. 
b, Heliocentric latitude, 
it, Geocentric longitude. 
(3, Geocentric latitude. 
6 or il, Longitude of ascending 

node. 
i, Inclination of orbit to the eclip- 
tic. 
unit w. Angular distance from perihe- 
lion to node. 
Distance from node, or argu- 
ment for latitude, 
(also ; a, Altitude. 
I A, Azimuth, 
's Equatorial radius. 



The Greek alphabet is here inserted to aid those who are not already 
familiar with it in reading the parts of the text in which its letters 



Letters. 


Names. 


A a 


Alpha 


B (36 


Beta 


T yC 


Gamma 


A 6 


Delta 


E e 


Epsilou 


ZC? 


Z5ta 


H n 


Eia 


e$Q 


Tbeta 


I i 


Iota 


Kk 


Kappa 


A*k 


Lambda 


UfA 


Mu 



Letters. 


Names. 


N v 


Nil 


2 £ 


Xi 


O o 


Otnicron 


J] ZS T 


Pi 


P PQ 


Rho 


2 a 5 


Sigma 


T t7 


Tau 


Tv 


Upsilon 


4> 


Phi 


x* 


Chi 


*T/> 


Psi 


Qa 


Omega 



THE METRIC SYSTEM. 

The metric system of weights and measures being employed iu 
this volume, the following relations between the units of this system 
most used and those of our ordinary one will be found convenient for 
reference : 

MEASURES OF LENGTH. 

1 kilometre = 1000 metres = 0-62137 mile. 
1 metre = the unit = 39-37 inches. 

1 millimetre = ygW ot? a metre = 0-03937 inch. 



MEASURES OF WEIGHT. 

1 millier or tonneau = 1,000,000 grammes = 2204*6 pounds. 
1 kilogramme = 1000 grammes = 2-2046 pounds. 

1 gramme = the unit = 15*432 grains. 

1 milligramme = -nfou- of a gramme = 0-01543 grain. 



The following rough approximations may be memorized : 

The kilometre is a little more than •& of a mile, but less than f of 
a mile. 

The mile is 1-ny kilometres. 

The kilogramme is 2} pounds. 

The pound is less than half a kilogramme. 



CHAPTER I. 

THE KELATION OF THE EAETH TO THE 
HEAYENS. 

§ 1. THE EARTH. 



The following are fundamental propositions of modern 
astronomy : 

I. The earth is approximately a sphere. — Besides the 
proofs of this proposition familiar 
to the student of geography, we 
have the fact that portions of the 
earth's surface visible from ele- 
vated positions appear to be 
bounded by circles. This property 
belongs only to the surface of a 
sphere. 

II. The directions which we call 
up and down are not invariable, 
hut are always toward or from the 
centre of the earth. — Therefore, 
they are different at different points 
of the earth's surface. 

HI. The earth is completely isolated in space. — The 
most obvious proof of this is that men have visited nearly 
every part of its surface without finding any communica- 
tion with other bodies. 

IV. The earth is one of a vast number of globular 
bodies, familiarly known as stars and planets, moving 
according to certain laws and separated by distances 




Fig. 2. 

Illustrating: the fact that the 
portions of ih>" earth visible 
from elevated positions, 3, $', 
S", etc., are bounded by circles. 



10 ASTRONOMY. 

■so immense, that the magnitudes of the bodies themselves 

are insignificant in comparison. 

The first conception the student of astronomy lias to 
form is that of living on the surface of a spherical earth, 
which, although it seems of immense size to him. is really 
but a point in comparison with the distances which sepa- 
rate him from the stars which he sees in the heavens. 

S 2. THE CELESTIAL SFHERE. 

The directions of the heavenly bodies are denned by 
their positions on an imaginary sphere called the celestial 
sphere. 

The celestial sphere is an imaginary hollow sphere, hav- 
ing the earth in its centre, and of dimensions so o-reat that 
the earth may be considered a point in comparison. 

One half of the celestial sphere is represented by the 
vault above our heads, commonly called the sky. in which 
the heavenly bodies appear to be set. This vault is called 
the visiole hemisphere^ and is bounded on all sides by the 
horizon. To complete the sphere it is supposed to extend 
below the horizon on all sides, our view of it being cut 
off by the earth on which we stand. The hemisphere in- 
visible to us is visible to those upon the opposite side of 
the earth. We may imagine a complete view of the 
sphere to be obtained by travelling around the earth. 

The celestial sphere being imaginary, may be supposed 
to have dimensions as great as we please. Convenience is 
gained by supposing it so large as to include all the heavenly 
bodies within it. The latter will then appear as if upon 
its interior surface, as shown in Fig. 3. Here the observer 
is supposed to be stationed in the centre 0. and to have 
around him the bodies />. q. r. s, t, etc. The sphere itself 
being supposed to extend outside of all these bodies, we 
may imagine lines drawn from the centre through each of 
them, directly away from the observer, until they inter- 
sect the sphere in the points P Q Ii S T. etc. These 



THE CELESTIAL SPHERE. 11 

latter points will represent the apparent positions of the 
bodies, as seen by the observer at 0. 




Fig. 3.— stars seen on the celestial sphere. 

If several of the bodies, as those marked t, t', t", are in 
the same straight line from the observer, they will appear 
as one body, and will be projected on the same point 
of the sphere. Hence positions on the celestial sphere 
represent the directions of the heavenly bodies from the 
observer, but not their distances. 

To fix the apparent positions of the heavenly bodies on 
the celestial sphere, certain circles are supposed io be 
drawn upon it, to which these positions are referred. 

The following propositions flow from the doctrine of 
the sphere. In Figs. 1 and 3, suppose the earth to be at 
O, and the circles to represent the outlines of the celestial 
sphere, then : 

I. Every straight line through the earth, when produced 
indefinitely, intersects the celestial sphere in two opposite 
points. 



12 ASTRONOMY. 

Since the earth is supposed to be a point in comparison 
with the sphere, the points in which a line intersects the 
sphere may be supposed opposite, whether the line passes 
through the centre or the surface of the earth. 

II. Every plane through the earth intersects the sphere 
in a great circle. 

III. For every such plane there is one line through the 
centre of the earth intersecting the plane at right angles. 
This line meets the sphere at the poles of the great circle in 
which the plane intersects the sphere. 

Example. — P P\ Fig. 1, is the line through perpen- 
dicular to the plane A B. P and P' are the poles of A B. 

IV. Every line through the centre has one plane per- 
pendicular to it, which plane intersects the sphere in the 
great circle whose poles are the intersections of the line 
with the sphere. 

Example. — The line Q Q' has the one plane CD 
through perpendicular to it. 

§ 3. RELATION OF THE SPHERE TO THE HORI- 
ZON—PLANE OP THE HORIZON. 

A level plane touching the spherical earth at the point 
where an observer stands is called the plane of the horizon. 

If we imagine this plane extended out indefinitely on 
all sides so as to reach the celestial sphere, it will inter- 
sect the latter in a great circle, called the celestial horizon. 
The celestial horizon is therefore the boundary between 
the visible and invisible hemispheres, the view of the lat- 
ter being cut off by the earth. 

We may also imagine a plane passing through the cen- 
tre of the earth, parallel to the horizon of the observer. 
This plane will intersect the celestial sphere in a circle 
below that of the observer's horizon by an amount equal 
to the radius of the earth. This circle is called the 
rational horizon, while that first defined is called the 
sensible horizon. But when the celestial sphere is con- 
sidered so immense that the earth may be regarded as a 



RELATIONS OF THE HORIZON. 



13 



point in comparison, the rational and sensible horizons are 
considered to coincide on the celestial sphere. 

The vertical line. — The vertical of any observer is the 
direction of a plumb line where he stands. Considering 
the earth as a perfect sphere, this 
direction is that from the ob- 
server to its centre, and is neces- 
sarily perpendicular to the plane 
of the horizon. If we consider 
the vertical to extend indefinite- 
ly in both directions, it will cut 
the celestial sphere in two oppo- 
site points. 

The zenith is that point of 
the celestial sphere in which 
the vertical intersects it above 
the observer. 

The nadir is that point in which the vertical intersects 
the celestial sphere directly below the observer's feet. 
The zenith and nadir are the two poles of the horizon. 

Vertical planes and circles. — A vertical plane is any 
plane which contains the vertical line of the observer, 
^ny vertical plane, when produced indefinitely, intersects 
the celestial sphere in a great circle passing through the 
zenith and nadir, and cutting the horizon at right angles. 

Such a great circle is called a vertical circle of the 
celestial sphere. 




Fig. 4. 



The Visiblt 



Hemisphere— S N, the 
horizon ; Z, the zenith ; S, any star ; 
S H, its altitude ; M N, its azimuth. 



A vertical circle may pass through any point on the 
celestial sphere. It is then called the vertical circle of 
that point. Hence we may imagine, passing through 
every star, a vertical circle extending from that star to the 
horizon and meeting the latter at a right angle. 

The altitude of a heavenly body is its elevation above 
the horizon, measured on its vertical circle, and expressed 
in degrees. 

The altitude of a body in the zenith is 90° ; half way 
between the horizon and zenith, it is 45° ; in the horizon 
it is 0° or zero. 



14 ASTRONOMY. 

When a body is below the horizon, and therefore invisi- 
ble, its altitude is considered to be algebraically negative. 

The zenith distance of a heavenly body is its angular 
distance from the zenith of the observer. It follows from 
the definitions that the altitude and zenith distance of a 
body together make 90°. That is, if a be the altitude and 
z the zenith distance : 

a + z = 90°. 
z = 90° — a. 

The azimuth of a star is the angular distance of the 
point where its vertical circle meets the horizon, from the 
north or south point of the horizon, expressed in degrees. 

Example. — In Fig. 4 N H is the azimuth of the star/& 

The azimuth of a body exactly in the east or west is 90°. 

The prime vertical is the vertical circle passing through 
the east and west points of the horizon. 

Co ordinates of a heavenly body. — The position of any 
heavenly body relative to the horizon of the observer is 
completely expressed by its altitude and azimuth. If, for 
example, we are told that the azimuth of a star is 20° from 
north to west, and its altitude 30°, we measure an arc of 
20° from the north point of the horizon toward west, and 
then direct our attention to the point 30° upward. This 
point, and this alone, will represent the position of the 
star upon the celestial sphere. 

Numbers or quantities which exactly define the position 
of a body, are called its co-ordinates. Hence, altitude 
and azimuth are a pair of co-ordinates which give the posi- 
tion of the body relative to the horizon. 

It must, however, be remembered, as already stated, 
that these two co-ordinates give only the direction, and 
not the distance, of a body. Knowing the direction of a 
body, we know where to look for it, or on what point of 
the celestial sphere it appears projected ; but we have no 
knowledge of its distance. When we know the latter, 
we have completely defined the position of the body in 
space. We, therefore, reach the following conclusions : 



THE DIURNAL MOTION. 15 

Three co-ordinates are necessary to fix the position of a 
hody in space. 

But two co-ordinates suffice to determine its apparent 
position on the celestial sphere. 

Horizons of different places. — Since the earth is spheri- 
cal in form and the horizon is a plane touching this sphere, 
every different place must have a different horizon. This 
fact of observation afforded the ancients their proof of 
the rotundity of the earth. It was found that an eclipse 
of the moon, which was seen at sunset in one place, oc- 
curred long after sunset at a place farther east. 1 It was 
also found that in travelling north, stars disappeared in 
the south horizon and rose above the north horizon. Thus 
the spherical form of the earth was known before the 
beginning of our era. 

Not only does the horizon change as an observer moves 
from place to place, but the horizon of any one place is 
continually changing in consequence of the earth's rota- 
tion on its axis. Hence, the altitude and azimuth of the 
heavenly bodies are continually changing, and no one 
altitude or azimuth is true except for a particular moment 
and for a particular place. Other circles of reference 
must, therefore, be sought when we desire to make use of 
co-ordinates which shall be permanent. 

§ 4. THE DIURNAL MOTION. 

The diurnal motion is that apparent motion of the sun, 
moon, and stars from east to west, in consequence of 
which they rise and set. 

The term diurnal is applied because the motion repeats 
itself day after day. v 

The diurnal motion is caused by a daily revolution of 
the earth on an axis passing through its centre, called the 
axis of the earth. This axis intersects the earth at two 
opposite poles called the north and south poles of the 
earth. 

If the earth's axis be continued indefinitely in both diree- 



16 ASTRONOMY. 

tions, it intersects the celestial sphere in two opposite 
points, called celestial poles. 

The north celestial pole corresponds to the north end of 
the earth's axis; the south celestial pole to the south end. 




Fig. 5. 

The Earth's Axis and the Plane of the Equator — N P, the North Pole; S P, 
the Sonth Pole ; E Q, plane of the Equator ; e q, the terrestrial Equator. 

The plane E Q, passing through the centre of the earth 
at right angles to the axis, is called the plane of the 
equator. 

The plane of the equator intersects the surface of the 
earth in a circle e q, called the terrestrial or geographical 
equator, passing through certain countries and oceans, as 
taught in geography. 

When the plane of the equator is continued out in- 
definitely so as to reach the celestial sphere it meets the 
latter in a great circle, called the celestial equator. 

The celestial equator is everywhere half way between 
the two celestial poles and 90° from each. The celestial 
poles are, therefore, the poles of the celestial equator. 



THE DIURNAL MOTION. 17 

Apparent diurnal motion of tlte celestial sphere. — 
The observer on the earth bein^ unconscious of its revo- 
lution, the celestial sphere appears to him to revolve in an 
opposite direction around the earth, while the latter ap- 
pears to remain at rest. The case is much the same as 
when we are on a steamer turning round, and, being un- 
conscious of the motion, the harbor, ships, and houses 
seem to be revolving in the opposite direction. 

So far as appearances are concerned, it is indifferent 
whether we conceive the earth or the heavens to revolve, 
and for the purpose of description it is easier to speak as 
if the motion were in the heavens. It must, however, 
be remembered, that the revolution of the celestial sphere 
is only apparent, the real motion being that of the earth. 

Since the diurnal motion is an apparent rotation of the 
celestial sphere around a fixed axis, it follows that there 
must be two points on this sphere where there is no mo- 
tion, namely, the celestial poles. Moreover, since the 
celestial poles are two opposite points, one pole must be 
above the horizon, and therefore on a visible point of this 
sphere, and the other pole below it and therefore invisible. 

The celestial pole visible to us is the northern one. To 
find it, let the reader look at the northern heavens, as rep- 
resented in Fig. 6, on any clear evening. The first star to 
be found is Polaris, or the Pole Star. It may be recog- 
nized by the Pointers, two stars in the constellation 
Ursa Major, familiarly known as the Great Dipper. The 
straight line through these stars, represented by the dotted 
line in the figure, passes near Polaris. 

Polaris is about 1J° from the pole. There is no visi- 
ble star exactly at the pole. 

The altitude of the pole above the horizon is different 
in different places, being equal to the latitude of the 
place. In most regions of the United States, the latitude 
is between 35° and 45°. 

The angular distance of a star from the north pole is 
called its north polar distance. 



18 ASTRONOMY. 

The following laws of the diurnal motion will now be 
clear. 

I. Every star in the heavens appears to describe a circle 
around the pole as a centre. 




Fig. 6. — the apparent diuknal motion. 

II. The greater the polar distance of the star the larger 
the circle. 

If the north polar distance is less than the altitude of 
the pole, the circle which the star describes will not meet 
the horizon at all, and the star will therefore neither rise 
nor set, but will simply perform an apparent diurnal revo- 
lution around the pole. Below the pole it will appear to 
move from west to east, gradually rising up in the north- 
east and passing toward the west, above the pole. The 
direction of the motion is shown by the arrows on Fig. 6. 



THE DIURNAL MOTION. 



19 



The circle within which the stars neither rise nor set is 
called the circle of perpetual apparition. The radius of 
the circle of perpetual apparition is equal to the altitude 
of the pole above the horizon. 

As a result of this apparent motion, each individual 
constellation changes its configuration with respect to the 
horizon, that part which is highest when the constellation 
is above the pole being lowest when below it. This is 
shown in Fig. 7, which represents a supposed constellation, 
at different times of the night, as it revolves around the 
pole. The same thing may be seen by simply turning 
Pig. 6 around and viewing it with different sides up. 




NORTH 

Fig. 7. 

If the polar distance of the star exceeds the altitude of 
the pole, it will dip below the horizon during a part of 
its diurnal course, and will be longer below it the greater 
its polar distance 

A star whose polar distance is 90° lies on the celestial 
equator, and one half its diurnal motion is below and the 
other half above the horizon. The sun is in the celestial 
equator about March 21st and September 21st, of each 
year, so that at these times the days and nights are of 
•equal length. 



20 ASTRONOMY. 

Looking farther south at the celestial sphere, we shall 
at length see stars which rise a little to the east of south, 
and set a little to the west, being above the horizon but a 
short time. 

The south pole is as far below our horizon as the north 
pole is above it. Hence, stars near the south pole never 
rise in our latitudes. The circle within which stars never 
rise is called the circle of perpetual disappearance. 

The meridian. — The plane of the meridian is that ver- 
tical plane which contains the earth's axis, and therefore 
passes through the zenith and pole, and through the earth's 
centre. 

The terrestrial meridian is the line in which the plane 
of the meridian intersects the surface of the earth. It is- 
a north and south line passing through the point where 
we suppose the observer to be situated. It follows that 
if several observers are north and south of each other they 
have the same meridian ; otherwise they have different 
meridians. 

The celestial meridian is the great circle in which the 
meridian plane cuts the celestial sphere. It passes from 
the north point of the horizon to the pole, thence 
through the zenith and south horizon, the nadir, and up 
to the north horizon again. 

The complete circle forming the celestial meridian is 
sometimes divided into two semicircles by the poles of 
the earth. That semicircle which passes from one pole 
to the other through the zenith is called the upper meri- 
dian ; that through the nadir is called the lower meri- 
dian. 

Terrestrial meridians are considered as belonging to the 
places they pass through. Thus, we speak of the meridian 
of Greenwich, or the meridian of Washington, meaning 
thereby that north and south semicircle on the earth's 
surface passing from one pole to the other through the 
Royal Observatory, Greenwich, or through the Naval 
Observatory, Washington. 



DIURNAL MOTION IN DIFFERENT LATITUDES. 21 



$ 5. 



THE DIURNAL MOTION IN DIFFERENT LATI- 
TUDES. 



As we have seen, the celestial horizon of an observer 
will change its place on the celestial sphere as the observer 
travels from place to place on the surface of the earth. 
If he moves directly toward the north his zenith will ap- 
proach the north pole, but as the zenith is not a visible 
point, the motion will be naturally attributed to the pole, 
which will seem to approach the point overhead. The 
new apparent position of the pole will change the aspect 
of the observer's sky, as the higher the pole appears above 
the horizon the greater the circle of perpetual apparition, 
and therefore the greater the number of stars, which 
never set. 




Fig. 8. — the parallel sphere. 

If the observer is at the north pole his zenith and the 
pole itself will coincide : half of the stars only will be vis- 
ible, and these will never rise or set, but appear to move 
around in circles parallel to the horizon. The horizon 
and equator will coincide. The meridian will be indeter- 
minate since Z and P coincide ; there will be no east and 
west line, and no direction but south. The sphere in this 
case is called a parallel sphere. 



22 ASTRONOMY. 

If instead of travelling to the north the observer should 
go toward the equator, the north pole would seem to ap- 
proach his horizon. When he reached the equator both 
poles would be in the horizon, one north and the other 
south. All the stars in succession would then be visible, 
and each would be an equal time above and below the 
horizon. 




Fig. 9. — the right sphere. 

The sphere, in this case is called a right sphere, because 
the diurnal motion is at right angles to the horizon. If now 
the observer travels southward from the equator, the south 
pole will become elevated above his horizon, and in the 
southern hemisphere ajmearances will be reproduced 
which we have already described for the northern, except 
that the direction of the motion will, in one respect, be 
different. The heavenly bodies will still rise in the east 
and set in the west, but those near the equator will pass 
north of the zenith instead of south of it, as in our lati- 
tudes. The sun, instead of movnng from left to right, 
there moves from right to left. The bounding line be- 
tween the two directions of motion is the equator, where 
the sun culminates north of the zenith from March till 
September, and south of it from September till March. 

If the observer travels west or east, the character 
of the diurnal motion will not change. 



CIRCLES OF THE SPHERE. 



23 



§ 6. CORRESPONDENCE OF THE TERRESTRIAL 
AND CELESTIAL SPHERES. 

Fundamental proposition. — The altitude of the pole 
above the horizon is equal to the latitude of the place. 

This may be shown as follows : 

Let L be a place on the earth PEpQ, Pp being the 
earth's axis, and E Q its equator, Z is the zenith, and H 
R the horizon of L. L Q 
is the latitude of L accord- 
ing to ordinary geographical 
definitions : i.e., it is its an- 
gular distance from the equa- 
tor. 

Prolong O P indefinitely 
to P' and draw LP" par- 
allel to it. To an observer at 
Z, the elevated pole of the 
heavens will be seen along 
the line L P" ', because at an 
infinite distance the distance 
P P" will appear like a 
point. II LZ =POQ, and 
P"LII=LOQ — that is, the elevation of the pole above 
the celestial horizon L H is equal to the latitude of the 
place as stated. 

Correspondence of 'zeniths. — The zenith of any point on 
the surface of the earth is considered as a corresponding 
point of the celestial sphere. If for a moment we suppose 
both spheres at rest, an observer travelling over the earth 
would find his zenith to mark out a path on the celestial 
sphere corresponding to his path on the earth. To un- 
derstand the relation, we may imagine that the observer's 
zenith is marked out by an infinitely long pencil, extending 
vertically above his head to the celestial sphere. 

Let Fig. 11 represent the celestial sphere, with the earth 
m the centre. 




Fig. 10. 
ZLP" = ZOP\ hence 



24 ASTRONOMY. 

At either pole of the earth (s or n), the vertical line 
Will extend to the celestial pole, JY P or £ jP, and the 




Fig. 11. 

Correspondence of Circles on the Celestial and Terrestrial Spheres. 

zenith will remain the same from day to day, because 
the position of the observer is not changed by the rota- 
tion of the earth. 

If the observer is in latitude 45°, he will in twenty-four 
hours, by the rotation of the earth, be carried around on a 
parallel of terrestrial latitude, 45° from the north pole of 
the earth. His zenith will, during the same time, describe 
a circle M L on the celestial sphere, corresponding to 
this parallel of latitude on the earth ; that is, a circle 45° 
from the celestial pole and 45° from the celestial equator. 

Next, let us suppose the observer on the earth's equator 
at e or q. His zenith will then be 90° from each pole. As 
the earth revolves on its axis, his zenith will describe a 



CIRCLES OF THE SPHERE. 25 

great circle, E ' Q, around the celestial sphere. This circle 
is, in fact, the celestial equator. The line from the centre 
of the earth through the observer to his zenith will de- 
scribe a plane, namely, the plane of the equator. (Cf. 
Fig. 5.) 

An observer in 45° south latitude, will be half way be- 
tween the equator and the south pole. By the diurnal 
motion of the earth, he will be carried around on the paral- 
lel of 45° south terrestrial latitude. Since his zenith is con- 
tinually 45° from the pole, it will describe a circle, SO, in 
the celestial sphere of 45° south polar distance. 

Thus, for each parallel of latitude on the earth, we 
have a corresponding circle on the celestial sphere, having 
its pole at the celestial pole. Let us now inquire how 
far the same thing is true of the meridians. The relation 
of the meridians is complicated by the earth's rotation, in 
consequence of which the celestial meridian of any place 
is continually in motion from west to east on the celestial 
sphere. To express the same thing in another form, the 
celestial sphere is apparently in motion from east to west, 
across the terrestrial meridian, the latter, it will be re- 
membered, remaining at rest relative to any given place 
on the earth. A north and south wall on the earth is al- 
ways iii the common plane of the terrestrial and celestial 
meridians of the place where it stands. 

Suppose now that we could by a sweep of a pencil in a 
moment mark out the semicircle of our meridian upon 
the heavens by a line from the north pole through the 
zenith and south horizon, to the ' south pole. At the end 
of an hour this semicircle would have apparently moved 
15° toward the west by the diurnal motion. Then im- 
agine that we again mark our meridian on the celestial 
sphere. The two semicircles will meet at each pole and 
be widest apart at the equator. Continuing the process 
for twenty-four hours, we should have twenty-tour semi- 
circles, all diverging from one pole and meeting at the 



26 ASTRONOMY. 

other, as shown in Fig. 11. The circles thus formed are 
called hour circles. Hence the definition : 

Hour circles on the celestial sphere are circles passing 
through the two poles and therefore cutting the equator 
at right angles. 

The hour circle of any particular star is the hour circle 
passing through that star. In Figure 11a let the outline 
represent the celestial sphere ; Z being the zenith and P 




Fig. 11a.— circles of the sphere. 

the north pole. Let A be the position of a star. 
Then PAB is, by definition, part of the hour circle of 
the star A. The angle ZPA is then called the hour 
angle of the star. Hence the definition : 

The hour angle of a star is the angle which its hour 
circle makes with the meridian of the place. 

The hour angle of a star is, therefore, continually chang- 
ing in consequence of the diurnal motion. 

The declination of a star is its distance from the equa- 
tor north or south. Thus, in the figure, CWJDE is 
the celestial equator and the arc B A is the declination of 



RIGHT ASCENSION AND DECLINATION. 27 

the star A. By the previous definition, PA is the polar 
distance of the star. Because P B is 90°, it follows that 
the sum of the polar distance and declination of a star 
make 90°. 

Therefore, if we put p, the polar distance of the star, 
and <5 its declination, we shall have : 

p + & = 90°. 

The declination and hour angle of a star are two co 
ordinates which completely define its position. 

From Figure 11a it can at once be seen that the lati- 
tude of a place on the earth's surface is equal to the 
declination of the zenith of that place, since the declina- 
tion of the zenith is equal to the altitude of the elevated 
pole. 

§ 7. RIGHT ASCENSION AND DECLINATION. 

Since the hour angle of a heavenly body is continually 
changing in consequence of the diurnal motion, it is 
necessary to have fixed circles on the sphere to which we 
may refer the position of a star. The circles already de- 
scribed will enable us to do this. 

We call to mind that to determine the longitude of a 
place, we choose some meridian, that of Greenwich or 
Washington, for instance, as a first meridian, and then de- 
fine the longitude of the place as the angle which its 
meridian makes with the first meridian. To define the 
corresponding co-ordinate of a fixed star, we choose a cer- 
tain hour circle on the celestial sphere as a standard, and 
express a celestial longitude of any heavenly body by the 
angle which its hour circle makes with this standard hour 
circle. This co-ordinate in the heavens, however, is not 
called longitude, but right ascension. 

The hour circle chosen in the heavens is that of the 
vernal equinox. What the vernal equinox is will be do- 
scribed hereafter. For our present purpose nothing more 



28 ASTRONOM T. 

is necessary than to understand that a particular circle is 
arbitrarily chosen. Hence the definition : 

The right ascension of a heavenly body is the angle 
which its hour circle makes with the hour circle passing 
through the vernal equinox. 

Besides the right ascension, another co-ordinate is re- 
quired to fix the position of the star, and for this we take 
the declination or angular distance from the celestial equa- 
tor already described. The relation of these co-ordinates 
to terrestrial ones are : 

To latitude on the earth corresponds declination in the 
heavens. 

To longitude on the earth corresponds right ascension 
in the heavens. 

§ 8. RELATION OP TIME TO THE SPHERE. 

Different Kinds of Time.— We have seen (p. 17) that 
the earth rotates uniformly on its axis — that is, it turns 
through equal angles in equal intervals of time. This ro- 
tation can be used to measure intervals of time when 
once a unit of time is agreed upon. The most natural 
unit is a day. 

A sidereal day is the interval of time required for the 
earth to make one complete revolution on its axis. Or, 
what is the same thing, it is the interval of time between 
two consecutive transits of a star over the same meridian. 
The sidereal day is divided into 21 sidereal hours ; each 
hour is divided into 60 minutes ; each minute into 60 
seconds. 

In making one revolution, the earth turns through 360°, 
so that 

24 hours = 360° ; also 

1 hour = 15° ; 1° = 4 minutes ; 

1 minute = 15' ; 1' = 4 seconds ; 

1 second = 15" ; 1" = 0-066 . . . sec. 

The hour-angle of any star on the meridian of a place 
is zero (by definition p. 26). It is then at its transit or 
culmination. 



SIDEREAL TIME. 29 

As the earth rotates, the meridian moves away (east- 
wardly) from this star, whose hour-angle continually in- 
creases from 0° to 360°, or from hours to 24 hours. 
Sidereal time can then be directly measured by the hour- 
angle of any star in the heavens which is on the meridian 
at an instant we agree to call sidereal hours. When this 
star has an hour-angle of 90°, the sidereal time is 6 hours ; 
when the star has an hour-angle of 180° (and is again on 
the meridian, but invisible unless it is a ciicumpolar star) it 
is 12' hours ; when its hour-angle is 270° the sidereal time 
is 1% hours, and, finally, when the star reaches the upper 
meridian again, it is 24 hours or hours. See Fig. 11a 
where E C WDh the apparent diurnal path of a star 
in the equator. It is on the meridian at C. 

Instead of choosing a star as the determining point 
whose transit marks sidereal hours, it is found more con- 
venient to select that point in the sky from which the 
right ascensions of stars are counted — the vernal equinox — 
the point V in the figure. The fundamental theorem of si- 
dereal time is, the hour-angle of the vernal equinox or the 
sidereal time is equal to the right ascension of the meri- 
dian, that is O I r — V C. 

To avoid continual reference to the stars, we set a clock 
so that its hands shall mark hours minutes seconds 
at the transit of the vernal equinox, and regulate it so that 
its hour-hand revolves once in 24 sidereal hours. Such a 
clock is called a sidereal clock. 

Time measured by the hour-angle of the sun is called 
true or apparent solar time. An apparent solar day is 
the interval of time between two consecutive transits oi 
the sun over the upper meridian. The instant oi the 
transit of the sun over the meridian of any place is the 
ap>parent noon of that place, or local apparent noon. 

When the sun's hour-angle is 12 hours or 180°, it is 
local apparent midnight. 

The ordinary sun-dial marks apparent solar time. As 
a matter of fact, apparent solar days are not equal. The 



30 ASTRONOMY. 

reason for this is fully explained later (p. 258). Hence 
our clocks are not made to keep tliis kind of time, for if 
once set right they would sometimes lose and sometimes 
gain on such time. 

A modified kind of solar time is therefore used, called 
mean solar time. This is the time kept by ordinary watches 
and clocks. It is sometimes called civil time. Mean solar 
time is measured by the hour-angle of the mean sun, a 
fictitious body which is imagined to move uniformly in 
the heavens. The law according to which the mean sun 
is supposed to move enables us to compute its exact position 
in the heavens at any instant, and to define this position by 
the two co-ordinates right ascension and declination. Thus 
we know the position of this imaginary body just as we 
know the position of a star whose co-ordinates are given, 
and we may speak of its transit as if it were a bright ma- 
terial point in the sky. A mean solar clay is the interval 
of time between two consecutive transits of the mean sun 
over the upper meridian. Mean noon at any place on the 
earth is the instant of the mean sun's transit over the meri- 
dian of that place. Twelve hours after local mean noon 
is local mean midnight. The mean solar day is divided 
into 24 hours of 60 minutes each. Each minute of mean 
time contains 60 mean solar seconds. 

We Lave thus three kinds of time. They are alike in one point. 
Each is measured by the hour-angle of some body, real or assumed. 
The body chosen determines the kind of time, and the absolute length 
of the unit — the day. The simplest unit is that determined by the 
uniformly rotating earth — the sidereal day ; the most natural unit is 
that determined by the sun itself — the apparent solar day, which, 
however, is a variable unit ; the most convenient unit is the mean 
solar day. 

Comparative Lengths of the Mean Solar and Sidereal 
Day.— As a fact of observation, it is found that the sun 
appears to move from west to east among the stars, about 
1° daily, making a complete revolution around the sphere 
in a year. The reason of this will be explained later (p. 
101). 



SIDEREAL TIME. 31 

Hence an apparent solar day will be longer than a 
sidereal day. For suppose the sun to be at the vernal 
equinox exactly at sidereal noon (0 hours) of Washington 
time on March 21st — that is, the vernal equinox and the 
sun are both on the meridian of Washington at the same 
instant. In 21 sidereal hours the vernal equinox will 
again be on the same meridian, but the sun will have 
moved eastwardly by about a degree, and the earth will 
have to turn through this angle and a little more in order 
that the sun shall again be on the Washington meridian, 
or in order that it may be apparent noon on March 22d. 
For the meridian to overtake the sun requires about 4 
minutes of sidereal time. The true sun does not move, as 
we have said, uniformly. The mean sun is supposed to 
move uniformly, but to make the circuit of the heavens 
in the same time as the real sun. Hence a mean solar day 
will also be longer than a sidereal day, for the same reason 
that the apparent solar day is longer. The exact rela- 
tion is : 

1 sidereal day = 0-997 mean solar day, 

24 sidereal hours = 23 h 56 m 4 s -091 mean solar time, 

1 mean solar day = 1-003 sidereal days, 

24 mean solar hours = 24 h 3 m 56 3 -555 sidereal time, 

and 

366-24222 sidereal days = 365-24222 mean solar days. 

Local Time.— When the mean sun is on the meridian of 
a place, as Boston, it is mean noon at Boston. When the 
mean sun is on the meridian of St. Louis, it is mean noon 
at St. Louis. St. Louis being west of Boston, and the 
earth rotating from west to east, the local noon of Boston 
occurs before the local noon at St. Louis. In the 
same way the local sidereal time at Boston at any given 
instant is expressed by a larger number than the local 
sidereal time of St. Louis at that instant. 

The sidereal time of our common noon is given in the 
astronomical ephemeris for every day of the year. It can 
be found within ten or twelve minutes at any time by re- 
membering that on March 2 1st it is sidereal hours about 



32 ASTRONOMY. 

noon, on April 21st it is about 2 hours sidereal time at 
noon, and so on through the year. Thus, by adding two 
hours for each month, and four minutes for each day after 
the 21st day last preceding, we have the sidereal time at 
the noon we require. Adding to it the number of hours 
since noon, and one minute more for every fourth of a day 
on account of the constant gain of tbe clock, we have the 
sidereal time at any moment. 

Example. — Find the sidereal time on July 4th, 1881,, 
at 4 o'clock a.m. We have : 

h m 

June 21st, 3 months after March 21st ; to be X 2, 6 
July 3d, 12 days after June 21st ; X 4, 48 

4 a.m., 16 hours after noon, nearly J of a day, 16 3 



22 51 



This result is within a minute of the truth. 



Relation of Time and Longitude. — Considering our civil 
time which depends on the sun, it will be seen that it ia 
noon at any and every place on the earth when the sun 
crosses the meridian of that place, or, to speak with more 
precision, when the meridian of the places passes under 
the sun. In the lapse of 24 hours, the rotation of the 
earth on its axis brings all its meridians under the sun in 
succession, or, which is the same thing, the sun appears to 
pass in succession all the meridians of the earth. Hence, 
noon continually travels westward at the rate of 15° in an 
hour, making the circuit of the earth in 24 hours. The 
difference between the time of day, or local time as it is 
called, at any two places, will be in proportion to the differ- 
ence of longitude, amounting to one hour for every 15 
degrees of longitude, four minutes for every degree, and 
so on. Vice versa, if at the same real moment of time 
we can determine the local times at two different places, 
the difference of these times, multiplied by 15, will give 
the difference of longitude. 



CHANGE OF DA Y. 33 

The longitudes of places are determined astronomically 
on this principle. Astronomers are, however, in the 
habit of expressing the longitude of places on the earth 
like the right ascensions of the heavenly bodies, not in 
degrees, but in hours. For instance, instead of saying 
that Washington is 77° 3' west of Greenwich, we com- 
monly say that it is 5 hours 8 minutes 12 seconds west, 
meaning that when it is noon at "Washington it is 5 hours 
8 minutes 12 seconds after noon at Greenwich. This 
course is adopted to prevent the trouble and confusion 
which might arise from constantly having to change hours 
into degrees, and the reverse. 

A question frequently asked in this connection is, 
Where does the day change ? It is, we will suppose, Sun- 
day noon at Washington. That noon travels all the way 
round the earth, and when it gets back to Washington 
again it is Monday. Where or when did it change from 
Sunday to Monday ? We answer, wherever people choose 
to make the change. Navigators make the change 
occur in longitude 180° from Greenwich. As this meri- 
dian lies in the Pacific Ocean, and scarcely meets any land 
through its course, it is very convenient for this purpose. 
If its use were universal, the day in question would be 
Sunday to all the inhabitants east of this line, and Mon- 
day to every one west of it. But in practice there have 
been some deviations. As a general rule, on those islands 
of the Pacific which are settled by men travelling east, 
the day would at first be called Monday, even though 
they might cross the meridian of 180°. Indeed the Rus- 
sian settlers carried their count into Alaska, so that when 
our people took possession of that territory they found 
that the inhabitants called the day Monday when they 
themselves called it Sunday. These deviations have, how- 
ever, almost entirely disappeared, and with few exceptions 
the day is changed by common consent in longitude 1S0 J 
from Greenwich. 









34 ASTRONOMY. 

S 9 DETERMINATIONS OF TERRESTRIAL LONGI- 
TUDES. 

We have remarked that, owing to the rotation of the earth, 
there is no such fixed correspondence between meridians on 
the earth and among the stars as there is between latitude on 
the earth and declination in the heavens. The observer 
can always determine his latitude by finding the declination 
of his zenith, but he cannot find his longitude from the 
right ascension of his zenith with the same facility, be- 
cause that right ascension is constantly changing. To deter- 
mine the longitude of a place, the element of time as mea- 
sured by the diurnal motion of the earth necessarily comes 
in. Let us once more consider the plane of the meridian 
of a place extended out to the celestial sphere so as to 
mark out on the latter the celestial meridian of the place. 
Consider two such places, Washington and San Francisco 
for example ; then there will be two such celestial meri- 
dians cutting the celestial sphere so as to make an angle of 
about forty-five degrees with each other in this case. Let 
the observer imagine himself at San Francisco. Then he 
may conceive the meridian of Washington to be visible 
on the celestial sphere, and to extend from the pole over 
toward his south-east horizon so as to pass at a distance of 
about forty-five degrees east of his own meridian. It 
would appear to him to be at rest, although really both 
his own meridian and that of Washington are moving in 
consequence of the earth's rotation. Apparently the stars 
in their course will first pass the meridian of Washington, 
and about three hours later will pass his own meridian. 
Now it is evident that if he can determine the interval 
which the star requires to pass from the meridian of Wash- 
ington to that of his own place, he will at once have the 
difference of longitude of the two places by simply turn- 
ing the interval in time into degrees at the rate of fifteen 
degrees to each hour. 

Essentially the same idea may perhaps be more readily 
grasped by considering the star as apparently passing over 



LONGITUDE. 35 

the successive terrestrial meridians on the surface of the 
earth, the earth being now supposed for a moment to be 
at rest. If we imagine a straight line drawn from the 
centre of the earth to a star, this line will in the course of 
twenty-four sidereal hours apparently make a complete 
revolution, passing in succession the meridians of all the 
places on the earth at the rate of fifteen degrees in an hour 
of sidereal time. If, then, Washington and San Francisco 
are forty-five degrees apart, any one star, no matter what 
its declination, will require three sidereal hours to pass 
from the meridian of Washington to that of San Francisco, 
and the sun will require three solar .hours for the same 
passage. 

Whichever idea we adopt, the result will be the same : 
difference of longitude is measured by the time required 
for a star to apparently pass from the meridian of one 
place to that of another. There is yet another way of 
defining what is in effect the same thing. The sidereal 
time of any place at any instant being the same with the 
right ascension of its meridian at that instant, it follows 
that at any instant the sidereal times of the two places will 
differ by the amount of the difference of longitude. For 
instance : suppose that a star in hours right ascension is 
crossing the meridian of Washington. Then it is hoars 
of local sidereal time at Washington. Three hours later 
the star will have reached the meridian of San Francisco. 
Then it will be hours local sidereal time at San Fran- 
cisco. Hence the difference of longitude of two places is 
measured by the difference of their sidereal times at the 
same instant of absolute time. Instead of sidereal times, 
we may equally well take mean times as measured by the 
sun. It being noon when the sun crosses the meridian of 
any place, and the sun requiring three hours to pass from 
the meridian of Washington to that of San Francisco, it 
follows that when it is noon at San Francisco it is three 
o'clock in the afternoon at Washington.* 

* The difference of longitude thus depends upon the angular fffc- 
tan ce of terrestrial meridian 8, and not upon the motion of a celestial body, 



36 ASTRONOMY. 

The whole problem of the determination of terrestrial 
longitudes is thus reduced to one of these two : either 
to find the moment of Greenwich or Washington time 
corresponding to some moment of time at the place 
which is to be determined, or to find the time required 
for the sun or a star to move from the meridian of Green- 
wich or Washington to that of the place. If it were 
possible to fire a gun every day at Washington noon 
which could be heard in an instant all over the earth, 
then observers everywhere, with instruments to deter- 
mine their local time by the sun or by the stars, would be 
able at once to fix their longitudes by noting the hour, 
minute, and second of local time at which the gun was 
heard. As a matter of fact, the time of Washington noon 
is daily sent by telegraph to many telegraph stations, and 
an observer at any such station who knows his local time 
can get a very close value of his longitude by observing the 
local time of the arrival of this signal. Human ingenuity 
has for several centuries been exercised in the effort to in- 
vent some practical way of accomplishing the equivalent 
of such a signal which could be used anywhere on the 
earth. The British Government long had a standing offer 
of a reward of ten thousand pounds to any person who 
would discover a practical method of determining the lon- 
gitude at sea with the necessary accuracy. This reward 
was at length divided between a mathematician who con- 
structed improved tables of the moon's motion and a 
mechanician who invented an improved chronometer. 
Before the invention of the telegraph the motion of the 
moon and the transportation of chronometers afforded 
almost the only practicable and widely extended methods 
of solving the problem in question. The invention of 
the telegraph offered a third, far more perfect in its appli- 

and hence the longitude of a place is the same whether expressed as a 
difference of two sidereal times or of two solar times. The longitude 
of Washington west from Greenwich is 5 h 8 m or 77°, and this is, in fact, 
the ratio of the angular distance of the meridian of "Washington from 
that of Greenwich to 360 D or 24 h . It is thus plain that the longitude is 
the difference of the simultaneous local times, whether solar or sidereaL 



LONGITUDE BY CHRONOMETERS 3? 

cation, but necessarily limited to places in telegraphic 
communication with each other. 

Longitude by Motion of the Moon. — When we de- 
scribe the motion of the moon, we shall see that it moves 
eastward among the stars at the rate of about thirteen de- 
grees per day, more or less. In other words, its right as- 
cension is constantly increasing at the rate of a degree in 
something less than two hours. If, then, its right ascension 
can be predicted in advance for each Jiour of Greenwich 
or Washington time, an observer at any point of the 
earth, by noting the local time at his station, when the 
moon has any given right ascension, can thence determine 
the corresponding moment of Greenwich time ; and hence, 
from the difference of the local times, the longitude of his 
place. The moon will thus serve the purpose of a sort of 
clock running on Greenwich time, upon the face of which 
any observer with the proper appliances can read the 
Greenwich hour. This method of determining longitudes 
has its difficulties and drawbacks. The motion of the 
moon is so slow that a very small change in its right ascen- 
sion will produce a comparatively large one in the Green- 
wich time deduced from it — about 27 times as great an 
error in the deduced longitudes as exists in the determi- 
nation of the moon's right ascension. With such instru- 
ments as an observer can easily carry from place to place, 
it is hardly possible to determine the moon's right ascen- 
sion within five seconds of arc ; and an error of this 
amount will produce an error of nine seconds in the 
Greenwich time, and hence of two miles or more in his 
deduced longitude. Besides, the mathematical processes 
of deducing from an observed right-ascension of the moon 
the corresponding Greenwich time are, under ordinary 
circumstances, too troublesome and laborious .to make this 
method of value to the navigator. 

Transportation of Chronometers. — The transportation 
of chronometers affords a simple and convenient, method 
of obtaining the time of the standard meridian at any 
moment. The observer sets his chronometer as nearly as 



38 ASTRONOMY. 

possible on Greenwich or Washington time, and deter- 
mines its correction and rate. This he can do at any sta- 
tion of which the longitude is correctly known, and at 
which the local time can be determined. Then, wherever 
he travels, he can read the time of his standard meridian 
from the face of his chronometer at any moment, and 
compare it with the local time determined with his transit 
instrument or sextant. The principal error to which this 
method is subject arises from the necessary uncertainty in 
the rate of even the best chronometers. This is the 
method almost universally iised at sea where the object is 
simply to get an approximate knowledge of the ship's 
position. 

The accuracy can, however, be increased by carrying a 
large number of chronometers, or by repeating the de- 
termination a number of times, and this method is often 
employed for fixing the longitudes of seaports, etc. 
Between the years 1848 and 1855, great numbers of chro- 
nometers were transported on the Cunard steamers plying 
between Boston and Liverpool, to determine the difference 
of longitude between Greenwich and the Cambridge Ob- 
servatory, Massachusetts. At Liverpool the chronometers 
were carefully compared with Greenwich time at a local 
observatory — that is, the astronomer at Liverpool found 
the error of the chronometer on its arrival in the ship, 
and then again when the ship was about to sail. When 
the chronometer reached Boston, in like manner its error 
on Cambridge time was determined, and the determination 
was repeated when the ship was about to return. Having 
a number of such determinations made alternately on the 
two sides of the Atlantic, the rates of the chronometers 
could be determined for each double voyage, and thus the 
error on Greenwich time could be calculated for the mo- 
ment of each Cambridge comparison, and the moment of 
Cambridge time for each Greenwich comparison. 

Longitude by the Electric Telegraph. — As soon as the 
electric telegraph was introduced it was seen by American 



LONGITUDE BY TELEGRAPH. 39 

astronomers that we here had a method of determining 
longitudes which for rapidity and convenience would 
supersede all others. The first application of this method 
was made in 1814: between Washington and Baltimore, 
under the direction of the late Admiral Charles Wilkes, 
U. S.N. During the next two years the method was intro- 
duced into the Coast Survey, and the difference of longitude 
between New York, Philadelphia, and Washington was 
thus determined, and since that time this method has had 
wide extension not only in the United States, but between 
America and Europe, in Europe itself, in the East and West 
Indies, and South America. The principle of the method 
is extremely simple. Each place, of which the difference of 
time (or longitude) is to be determined, is furnished with a 
transit instrument, a clock and a chronograph ; instruments 
described in the next chapter. Each clock is placed in 
galvanic communication not only with its own chronograph, 
but if necessary is so connected with the telegraph wires 
that it can record its own beat upon a chronograph at the 
other station. The observer, looking into the telescope 
and noting the crossing of the stars over the meridian, 
can, by his signals, record the instant of transit both on his 
own chronograph and on that of the other station. The 
plan of making a determination between Philadelphia and 
Washington, for instance, was essentially this : When 
some previously selected star reached the meridian at Phil- 
adelphia, the observer pointed his transit upon it, and as 
it crossed the wires, recorded the signal of time not only 
on his own chronograph, but on that at Washington. 
About eight minutes afterward the star reached the 
meridian at Washington, and there the observer recorded 
its transit both on his own chronograph and on that at 
Philadelphia. The interval between the transit over the 
two places, as measured by either sidereal clock, at once 
gave the difference of longitude. If the record was in- 
stantaneous at the two stations, this interval ought to be 
the same, whether read off the Philadelphia or the Wash- 



40 ASTRONOMY. 

ington chronograph. It was found, however, that there 
was a difference of a small fraction of a second, arising 
from the fact that electricity required an interval of time, 
minute but yet appreciable, to pass between the two 
cities. The Philadelphia record was a little too late in 
being recorded at Washington, and the Washington one a 
little too late in being recorded at Philadelphia. We 
may illustrate this by an example as follows : 

Suppose E to be a station one degree of longitude east 
of another station, W ; and that at each station there is a 
clock exactly regulated to the time of its own place, in 
which case the clock at E will be of course four minutes 
fast of the clock at W ; let us also suppose that a signal 
takes a quarter of a second to pass from one station to the 
other : 

Then if the observer at E sends a signal to W at exactly 

noon by his clock 12 h m 8 .00 

It will be received at W at ll h 56 ra 0\25 

Showing an apparent difference of time of 3 m 59 9 .75 

Then if the observer at W sends a signal at noon by his 

clock 12 h m s .0O 

It will be received at E at 12 h 4 m 9 .25 

Showing an apparent difference of time of 4 ra 3 .25 

One half the sum of these differences is four minutes, 
which is exactly the difference of time, or one degree of 
longitude ; and one half their difference is twenty-five 
hundredths of a second, the time taken by the electric im- 
pulse to traverse the wire and telegraph instruments. 

This is technically called the " wave and armature 
time." 

We have seen that if a signal could be made at Wash- 
ington noon, and observed by an observer anywhere sit- 
uated who knew the local time of his station, his longi- 
tude would thus become known. This principle is often 
employed in methods of determining longitude other than 
those named. For example, the instant of the beginning 



THEORY OF THE SPHERE. 41 

and ending of an eclipse of the sun (by the moon) is a 
perfectly definite phenomenon. If this is observed by 
two observers, and these instants noted by each in the 
local time of his station, then the difference of these 
local times (subject to small corrections, due to parallax, 
etc.) will be the difference of longitude of the two sta- 
tions. 

The satellites of Jupiter suffer eclipses frequently, and 
the Greenwich and Washington times of these phenomena 
are computed and set down in the Nautical Almanac. Ob- 
servations of these at any station will also give the differ- 
ence of longitude between this station and Greenwich or 
Washington. As, however, they require a larger tele- 
scope and a higher magnifying power than can be used at 
sea, this method is not a practical one for navigators. 



§ 10. MATHEMATICAL THEORY OF THE CELESTIAL 
SPHERE. 

In this explanation of the mathematical theory of the relations of 
the heavenly bodies to circles on the sphere, an acquaintance with 
spherical trigonometry on the part of the reader is necessarily pre- 
supposed. The general method by which the position of a point on 
the sphere is referred to fixed points or circles is as follows : 

A fundamental great circle E V Q, Fig. 12 is taken as a basis, 
and the first co-ordinate * of the body is its angular distance from 
-this circle. When the earth's equator is taken as the fundamental 
circle, this distance is on the earth's surface called Latitude ; on the 
celestial sphere the corresponding distance is called Bed in at ion. If 
the horizon is taken as the fundamental circle the distance is called 
Altitude. Altitude is therefore angular distance above the horizon. 
To distinguish between distances on opposite sides of the circle, dis- 
tances on one side are regarded as algebraically positive quantities, 
and on the other side as negative, Iu the case of the equator the 
north side, and in that of the horizon the upper side, are considered 
positive. Hence, if a body is below the horizon its altitude is nega- 
tive, and the latitude of a city south of the earth's equator is, in 
astronomical language, considered as negative. 

Instead of the co-ordinate we have described, another called zenith 
or polar distance is frequently employed. The fundamental circle is 

* The co-ordinates of a body are those measure*, whether of angles or linos, which 
define its position. For instance, the geographical co-ordinates of a city are its 

latitude and longitude. To fix a position on a sphere or Other surface two co-ordi- 
nates are necessary, while in space three are required. 






42 



ASTRONOMY. 




everywhere 90° from its positive pole, P. Hence, if A is the position 

of a star or other point on the 
sphere, and we put 

d, its declination or altitude, 

= a A. 

p, its polar or zenith distance 
=P A, we shall have 

6 + p= 90°, 
or, 

p=90° — 6. 

If the star is south of the 
fundamental circle, at B for ex- 
ample, 6 being negative^ will ex- 
ceed 90°. This quantity p may 
range from zero at the one pole 
Fid. 12. to 180° at the other, and will al- 

ways be algebraically positive. 
It is on this account to be preferred to <5, though less frequently 
used. 

II. The second co-ordinate required to fix a position on the celes- 
tial or terrestrial sphere is longitude right ascension, or azimuth, ac- 
cording to the fundamental plane adopted. It is expressed by the 
position of the great circle or meridian P A a P' which passes 
through the position from one pole to the other, at right angles to 
the fundamental circle. An arbitrary point, Ffor instance, is chosen 
on this latter circle, and the longitude is the angle V a, from this 
point to the intersection of the meridian or vertical circle pissing 
through the object. We may also consider it as the angle V P A 
which the circle passing through the object makes with the circle 
P V, because this angle is equal 
to Va. The angle is commonly 
counte 1 from V toward the right, 
and from 0° round to 3G0 D , so as 
to av>>id using negative angles. 
If the observer is stationed in 
the centre of the sphere, wi:h his 
head toward the positive pole P, 
the positive direction should be 
from right to left around the 
sphere. When the horizon is 
taken as the fundamental circle 
or plane, this secondary co-ordi- 
nate is called the azimuth, and 
should be counted from the south 
point toward east, or from the Fig. 13. 

north point toward west, but is 

commonly counted the other way. Tt may be defined as the angular 
distance of the vertical circle passing through the object from the 
south point of the horizon. 




THEORY OF THE SPHERE. 



43 



The hour angle of a star is measured by the interval which has 
elapsed, or the angle through which the earth has revolved on its 
axis, since the star crossed the meridian. In Fig. 13 Z being the 
zenith and P the pole, the angle Z P 8 is the hour angle of the star 
8. This angle is measured at the pole. If we put 

r, the sidereal time, 

or, the right ascension of the object, we shall have 

Hour angle, h = r — a. 

It will be negative before the object has passed the meridian, and 
positive afterward. It differs from right ascension only in the point 
from which it is reckoned, and the direction which is considered 
positive. The right ascension is measured toward the east from a 
point (the vernal equinox) which is fixed among the stars, while the 
hour angle is measured toward the west from the meridian of the 
observer, which meridian is constantly in motion, owing to the 
earth's rotation. 

We have next to show the trigonometrical relations which subsist 
between the hour angle, declination, altitude, and azimuth. Let 




Fig. 14. 



Fig. 14 be a view of the celestial hemisphere which is above the 
horizon, as seen from the east. We then have : 

HE R W, the horizon. 

P, the pole. 

Z, the zenith of the observer. 

H M Z P R, the meridian of the observer. 

P R, the latitude of the observer, which call <f>. 

Z P, = 90° — 0, the co-latitude. 

P S, the north polar distance of the star = 90° — decliuation. 

T 8, its altitude, which call a. 

Z *S, its zenith distance = 90° — a. 

M Z S, its azimuth, = 180° — angle 8 Z P. 
Z P 8, its hour angle, which call h. 

The spherical triangle Z P S, of which the angles are formed by 



44 ASTRONOMY. 

the zenith, the pole, and the star, is the fundamental triangle of our 
-problem. The latter, as commonly solved, may be put into two forms. 

I. Given the latitude of the place, the declination or polar dis- 
tance of the star, and its hour angle, to find its altitude and azimuth. 

"We have, by spherical trigonometry, considering the angles and 
sides of the triangle Z P 8 : 

cos Z S = cos P Z cos P S + sin P Z sin P S cos P. 
sin Z S cos Z = sin P Z cos PS — cos P Z sin P S cos P. 
sin Z S sin Z = sin P 8 sin P. 

By the above definitions, 

Z S — 90° — a, (a being the altitude of the star). 
P Z = 90° — (p, (<p being the latitude of the place). 
P S — 90° — 6, (6 being the declination of the star, + when north) 

P = h, the hour angle. 

Z = 180° — z, (z being the azimuth). 

Making these substitutions*, the equation becomes : 



sin a = 


sin <£ sin <5 + cos <f> cos <5 cos h. 


cos a cos z = 


— cos <}> sin 6 + sin <£ cos <5 cos A. 


cos a sin g = 


cos <5 sin ^. 



From these equations sin a and cos a may be obtained separately, 
and, if the computation is correct, they will give the same value of a. 
If the altitude only is wanted, it may be obtained from the first 
equation alone, which may be transformed in various ways, explained 
in works on trigonometry. 

II. Given the latitude of the place, the declination of a star, and 
its altitude above the horizon, to find its hour angle and (if its right 
ascension is known) the sidereal time when it had the given altitude. 

We find from the first of the above equations, 



cos h = 
or we may use: 



sin a — sin <j> sin d t 



sin 2 £ft = \ 



cos <p cos cJ 
cos (<p — 6) — sin a 



cos ty cos 6 

Having thus found h, we have 

Sidereal time = h + <?, 

a being the star's right ascension, and the hour angle h being changed 
into time by dividing by 15. 

III. An interesting form of this last problem arises when we sup- 
pose a = 0, which is the same thing as supposing the star to be in 



ASTRONOMY. 



45 



the horizon, and therefore to be rising or setting. The value of h 
will then be the hour angle at which it rises or sets ; or being 
changed to time by dividing by 15, it will be the interval of sidereal 
time between its rising and its passage over the meridian, or be- 
tween this passage and its setting. This interval is called the semi- 
diurnal arc, and by doubling it 
we have the time between the 
rising and setting of the star or 
other object. Putting a = in 
the preceding expression for cos 
h we find for the semi diurnal 
arc h, 



cos h = 



sin <p sin 6 
cos </> cos 6 
— tan <p tan 6, 




and the arc during which the 
star is above the horizon is 2 h. 
From this formula may be 
deduced at once many of the 
results given in the preceding -c, 1pr ■, 

o^f,*™* FlG. 15. — UPPER AND LOWER DITJR- 

secLions. VAT a -una 

(1). At the poles <p = 90% ^ N L RCS ' 

tan <p = infinity, and therefore cos h = infinity. But the cosine of 
an angle can never be greater than unity ; there is therefore no value 
of h which fulfils the condition. Hence, a star at the pole can 
neither rise nor set. 

(2). At the earth's equator d> = 0°, tan = 0, whence cos h = 0, 
h = 90°, and 2 h = 180°, whatever be 6. This being a semicircum- 
ferenceall the heavenly bodies are half the time above the horizon to 
an observer on the equator. 

. (3). If 5 = 0° (that is, if the star is on the celestial equator), then 
tan (5 = 0, and cos h = 0, h = 90°, 2 h = 180°, so that all stars on 
the equator are half the time above the horizon, whatever be the lati- 
tude of the observer. Here we except the pole, where, in this case, 
tan </> tan 6 = a x 0, an indeterminate quantity. In fact, a star on 
the celestial equator will, at the pole of the earth, seem to move round 
in the horizon. 

(4). The above value of cos h may be expressed in the form : 



cos h = — 



tan d 
cot 



tan (5 



tan (90 — o) 



This shows that when 8 lies outside the limits -f (90° — 6) and 
— (90° — 0), cos h will lie without the limits — 1 and + 1, and 
there will be no value of h to correspond. Hence, in this case, the 
stars neither rise nor set. These limits correspond to those of per- 
petual apparition and perpetual disappearance. 

(5). In the northern hemisphere <p and tan. <p are positive. Then, 
when 8 is positive, cos h is negative, and h > 90°, 2 h > 180". With 






46 



ASTRONOMY. 



negative A, cos h is positive, h < 90°, 2 h < 180°. Hence, in north- 
ern latitudes, a northern star is more than half of the time above the 
horizon, and a southern star less. In the southern hemisphere, <p and 
tan are negative, and the case is reversed. 

(6). If, in the preceding case, the declination of a body is supposed 
constant and north, then the greater we make <j> the greater the nega- 
tive value of cos h and the greater h itself will be. Considering, in 
succession, the cases of north and south decli iation and north and 
south latitude, we readily see that the farther we go to the north on 
the earth, the longer bodies of north declination remain above the 
horizon, and the more quickly those of south declination set. In the 
southern hemisphere the reverse is true. Thus, in the month of 
June, when the sun is north of the equator, the dnys are shortest 
near the south pole, and continually increase in length as we go north. 



Examples. 

(1). On April 9, 1879, at Washington, the altitude of Rigel above 
the west horizon was observed to be 12° 25'. Its position was: 

Right ascension = 5 h 8 m 44 s -27 = a. 

Declination = — 8° 20' 36" = <*. 

The latitude of Washington is + 38° 53' 39" = 4>. 

What was the hour angle of the star, and the sidereal time of ob- 
servation ? 

Ig sin a— 9 • 332478 

Ig sin <p = 9-797879 
lg sin d = — 9-161681 



— lg sin <j> sin 6 = 

— sin <f> sin 6 = 
sin a = 


8 





sin a — sin <j> sin 6 = 

lg cos <j> = 
U cos S = 




9 
9 



lg cos <p cos 6 
lg (sin a — sin <p sin «\) 



= 9 



959560 

091109 

215020 



306129 

891151 
995379 

886530 
485905 



lg cos h = 9-599375 

h = 
h -- 15 = 

a = 



sidereal time = 



16° 34' 33" 

4 h 26 ra 18 s .20 
5 h 8 m 44 5 .27 
9 h 35 m 2 S .47 



(2). Had the star been observed at the same altitude in the east, 
what would have been the sidereal time ? 
Ans. a — h = h 42 m 26 9 .07. 



DETERMINATION OF LATITUDE. 



47 



(3). At what sidereal time does Rigel rise, and at what sidereal 
time does it set in the latitude of Washington ? 
— tg = — 9-906728 
tg(5 = — 9-166301 



cos h 


= 


- 9 


073029 




h 


= 


S3 C 


12' 


19" 




-r- 15 


= 


5" 


32" 


49 s 


27 


a 


= 


5 h 


8 n 


44° 


27 



rises 23 h 35"'55\00 
sets 10 1 ' 41 ,u 33S54 

(4). What is the greatest altitude of Rigel above the horizon of 
Washington, and what is its greatest depression below it V Ans. 
Altitude=42" 45' 45" ; depression = 59 26' 57 . 

(5). What is the greatest altitude of a star on the equator in the 
meridian of Washington ? Ans. 51" 6' 21". 

(6). The declination of the pointer in the Great Bear which is 
nearest ihe pole is 62 ' 30' N., at what altitude does it pass above 
the pole at Washington, and at what altitude does it pass below it V 
Ans. 66° 23' 39" above the pole, and 11" 23' 39' when below it. 

(7). If the declination of a star is 50° N., what length of sidereal 
time is it above the horizon of Washington and what length below it 
during its apparent diurnal circuit? Ans. Above, 21 h 52 ,u ; below, 
2 U 8 m . 

§ 11. DETERMINATION OF LATITUDES ON THE 
EARTH BY ASTRONOMICAL OBSERVATIONS. 

Latitude from circumpolar stars. — In Fig. 16 let Z represent the 
zenith of the place of observation, P the pole, and HPZ R the me- 
ridian, the observer being at the 

centre of the sphere. Suppose 
-Sand IS' to be the two points 
at which a circumpolar star 
crosses the meridian in the de- 
scription of its apparent diurnal 
orbit. Then, since P is midway 
between S and S', 

ZS + ZS' 

7T = ZP = 90° - 6. 



or, 



Z+Z' 



= 90° — 



If, then, we can measure the dis- 
tances Z and Z >', we have 

Z+Z' 
$ = 90° ~- 




Fig. 16. 



which serves to determine <j>. The distances Z imd Z' can b< 






48 



ASTRONOMY. 



ured by the meridian circle or the sextant — both of which instru- 
ments are described in the next chapter — and the latitude is then 
known. Z and Z' must be freed from the effects of refraction. In 
this method no previous knowledge of the star's declination is re- 
quired, provided it remains constant between the upper and lower 
transit, which is the case for fixed stars. 

Latitude by Circum-zenith Observations. — If two stars 
£ and S', whose declinations 6 and 6' are known, cross the meridian, 
one north and the other south of the zenith, at zenith distances Z 8 

and Z8\ which call Z and Z\ and 
if we have measured Z and Z\ we 
can from such measures find the 
latitude ; f or = 6 + Z and = 
6' — Z\ whence 

9 = i[(6 + d , ) + (Z-Z% 

It will be noted that in this meth- 
od the latitude depends simply 
upon the mean of two declinations 
which can be determined before- 
hand, and only requires the differ- 
ence of zenith distances to be ac- 
curately measured, while the ab- 
solute values of these are unknown. In this consists its advantage. 
Latitude by a Single Altitude of a Star.— In the triangle 
ZTS (Fig. 14) the sides are ZP = 90° — 0; P 8 = 90° — <*; Z8 — 
Z= 90° — «; ZPS — h = the hour-angle. If we can measure at 
any known sidereal time 8 the altitude a of the star 8, and if we 
further know the right ascension, a, and the declination, <*, of the 
body (to be derived from the Nautical Almanac or a catalogue of 
stars), then we have from the triangle 

sin 6 + cos cos 6 cos h (1) 

is the only unknown. 




Fib. 17. 



sin a = sin 



a and S are known, and h = 6 — a, so that 

Put 

d sin D = sin 6 (2) 

d cos D = cos 6 cos A, (3) 

whence d and D are known, and (1) becomes 

d cos (0 — D) = sin «, (4) 

whence — D and are known. The altitude a is usually measured 
with a sextant. 

Latitude by a Meridian Altitude. — If the altitude of the 
body is observed on the meridian and south of the zenith, the equa- 
tion above becomes, since h = in this case, 

sin = sin a sin 6 + cos a cos 6 f 
or, 

sin = cos ( a — 6) .*. <p = 90° — a -f 6, 

which is evidently the simplest method of obtaining <p from a meas- 



PARALLAX. 



49 



ured altitude of a body of known declination. The last method is that 
commonly used at sea, the altitude being measured by the sextant, 
The student can deduce the formula for a north zenith-distance. 



§ 12. PABALLAX AND SEMIDIAMETEB. 

An observation of the apparent position of a heavenly 
body can give only the direction in which it lies from the 
station occupied by the observer without any direct indi- 
cation of the distance. It is evident that two observers 
stationed in different parts of the earth will not see such 
a body in the same direction. In Fig. 18, let S' be a sta- 




FlG. 18.— PARALLAX. 

tion on the earth, P a planet, Z the zenith of S', and the 
outer arc a part of the celestial sj>here. An observation 
of the apparent right ascension and declination of P taken 
from the station & will give us an apparent position P\ 
A similar observation at S" will give an apparent position 
Jr", while if seen from the centre of the earth the appar- 
ent position would be P r The angles P' P P / and 
P" P P n which represent the differences of direction, are 
called parallaxes. It is clear that the parallax of a body 
depends upon its distance from the earth, being greater 
the nearer it is to the earth. 

The word parallax having several distinct applications, 
we shall give them in order, commencing with the most 
general signification. 



50 ASTRONOMY. 

(1.) In its most general acceptation, parallax is the differ- 
ence between the directions of a body as seen from two 
different standpoints. This difference is evidently equal 
to the angle made between two lines, one drawn from each 
point of observation to the body. Thus in Fig. 18 the 
difference between the direction of the body P as seen 
from and from S' is equal to the angle P' P P n and this 
again is equal to its opposite angle 3' P C. This angle is, 
however, the angle between the two points C and S' as 
seen from P : we may therefore refer this most general 
definition of parallax to the body itself, and define parallax 
as the angle subtended by the line between two stations as 
seen from a heavenly body. 

(2.) In a more restricted sense, one of the two stations is 
supposed to be some centre of position from w T hich we 
imagine the bod} r to be viewed, and the parallax is the 
difference between the direction of the body from this 
centre and its direction from some other point. Thus 
the parallax of which we have just spoken is the differ- 
ence between the direction of the body as seen from the 
centre of the earth C and from a point on its surface as S f . 
If the observer at any station on the earth determines 
the exact direction of a body, the parallax of which we 
speak is the correction to be applied to that direction in 
order to reduce it to what it would have been had the ob- 
servation been made at the centre of the earth. Obser- 
vations made at different points on the earth's surface are 
compared by reducing them all to the centre of the earth. 

We may also suppose the point C to be the sun and the 
circle & S" to be the earth's orbit around it. The paral- 
lax will then be the difference between the directions of 
the body as seen from the earth and from the sun. This 
is termed the annual parallax, because, owing to the an- 
nual revolution of the earth, it goes through its period 
in a' year, always supposing the body observed to be at 
rest. 

(3.) A yet more restricted parallax is the horizontal 



PARALLAX. 



51 



parallax of a heavenly body. The parallax first described 
in the last paragraph varies with the position of the ob- 
server on the surface of the earth, and has its greatest 
value when the body is seen in the horizon of the ob- 
server, as may be seen by an inspection of Fig. 19, in 
which the angle GPS attains its maximum when the line 
P S is tangent to the earth's surface, in which case P 
will appear in the horizon of the observer at S. 




Fig. 19.— horizontal parallax. 

The horizontal parallax depends upon the distance of a 
body in the following manner: In the triangle G P S, 
right-angled at S, we have 

GS = CPsin GPS. 

If, then, we put 

p, the radius of the earth G S\ 

r, the distance of the body P from the centre of the 
earth ; 

7T, the angle SPG, or the horizontal parallax, 
we shall have, 

P 

p = r sin n ; r = — 

sin n 

Since the earth is not perfectly spherical, the quantity p 
is not absolutely constant for all parts of the earth, and its 
greatest value is usually taken as that to which the hori- 
zontal value shall be referred. This greatest value is, as 
we shall hereafter see, the radius of the equator, and the 



52 ASTRONOMY. 

corresponding value of the parallax is therefore called the- 
equatorial horizontal parallax. 

When the distance r of the body is known, the equa- 
torial horizontal parallax can be found by the first of the 
above equations ; when the parallax can be observed, the 
distance r is found from the second equation. How this 
is done will be described in treating the subject of celes- 
tial measurement. 

It is easily seen that the equatorial horizontal parallax, 
or the angle C P & is the same as the angular semi- 
diameter of the earth seen from the object P. In fact, 
if we draw the line P S' tangent to the earth at S\ the 
angle S P S' will be the apparent angular diameter of the 
earth as seen from TV and will also be double the angle 
OPS. The apparent semi-diameter of a heavenly body 
is therefore given by the same formulae as the parallax, 
its own radius being substituted for that of the earth. If 
we put, 

p, the radius of the body in linear measure ; 

r, the distance of its centre from the observer, expressed, 
in the same measure ; 

s, its angular semi-diameter, as seen by the observer ; 
we shall have, 

P 
sm s = — . 
r 

If we measure the semi-diameter s, and know the dis- 
tance, r, the radius of the body will be 

p = r sin s. 

Generally the angular semi-diameters of the heavenly 
bodies are so small that they may be considered the same 
as their sines. We may therefore say that the apparent 
angular diameter of a heavenly body varies inversely as- 
its distance. 



CHAPTER II. 

ASTRONOMICAL INSTRUMENTS. 

§ 1. THE REFRACTING TELESCOPE. 

In explaining the theory and use of the refracting tele« 
scope, we shall assume that the reader is acquainted with 
the fundamental principles of the refraction and disper- 
sion of light, so that the simple enumeration of them 
will recall them to his mind. These principles, so far 
as we have occasion to refer to them, are, that when 
a raj of light passing through a vacuum enters a trans- 
parent medium, it is refracted or hent from its course 
in a direction toward a line perpendicular to the sur- 
face at the point where the ray enters ; that this bend- 
ing follows a certain law known as the law of sines ; 
that when a pencil of rays emanating from a luminous 
point falls nearly perpendicularly upon a convex lens, 
the rays, after passing through it, all converge toward a 
point on the other side called a focus ; that light is com- 
pounded of rays of various degrees of refrangibility, so 
that, when thus refracted, the component rays pursue 
slightly different courses, and in passing through a lens 
come to slightly different foci ; and finally, that the ap- 
parent angular magnitude subtended by an object when 
viewed from any point is inversely proportional to its 
distance.* 



* More exnctly, in the case of a globft. the sine of the angle is in 
versely as the distance of the object, as shown on the preceding page. 



54 



ASTRONOMY. 



"We shall first describe the telescope in its simplest 
form, showing the principles upon which 
its action depends, leaving out of considera- 
tion the defects of aberration which require 
special devices in order to avoid them. In 
the simplest form in which we can conceive 
% of a telescope, it consists of two lenses of 
I unequal focal lengths. The purpose of one 
® of these lenses (called the objective, or object 
% glass) is to bring the rajs of light from a 
| distant object at which the telescope is 
pointed, to a focus and there to form an 
image of the object. The purpose of the 
other lens (called the eye-piece) is to view 
this object, or, more precisely, to form an- 
£ other enlarged image of it on the retina of 
the eye. 

The figure gives a representation of the 
course of one pencil of the rays which go to 
form the image A I' of an object I B after 
passing through the objective 0\ The 
pencil chosen is that composed of all the 
rays emanating from I which can possibly 
fall on the objective 0'. All these are, 
by the action of the objective, concentrated 
at the point A In the same way each point 
of the image out of the optical axis A B 
emits an oblique pencil of diverging rays 
which are made to converge to some point 
w . of the image by the lens. The image of 
£j the point B of the object is the point A of 
the image. We must conceive the image of 
any object in the focus of any lens (or 
mirror) to be formed by separate bundles 
of rays as in the figure. The image thus 
formed becomes, in its turn, an object to 
he viewed by the eye-piece. After the rays meet to form 



MAGNIFYING POWER OF TELESCOPE. 55 

the image of an object, as at T, they continue on their 
course, diverging from T as if the latter were a material 
object reflecting the light. There is, however, this excep- 
tion : that the rays, instead of diverging in every direction, 
only form a small cone having its vertex at /', and having 
its angle equal to T 0' . The reason of this is that 
only those rays which pass through the objective can form 
the image, and they must continue on their course in 
straight lines after forming the image. This image can 
now be viewed by a lens, or even by the unassisted eye, if 
the observer places himself behind it in the direction A y 
so that the pencil of rays shall enter his eye. For the pres- 
ent we may consider the eye-piece as a simple lens of 
short focus like a common hand-magnifier, a more com- 
plete description being given later. 

Magnifying Power.— To understand the manner in 
which the telescope magnifies, we remark that if an eye at 
the object-glass could view the image, it would appear of 
the same size as the actual object, the image and the object 
subtending the same angle, but lying in opposite direc- 
tions. This angular magnitude being the same, whatever 
the focal distance at which the image is formed, it follows 
that the size of the image varies directly as the focal length 
of the object-glass. But when we view an object with a 
lens of small focal distance, its apparent magnitude is the 
same as if it were seen at that focal distance. Consequently 
the apparent angular magnitude will be inversely as the 
focal distance of the lens. Hence the focal image as 
seen with the eye-piece will appear larger than it would 
when viewed from the objective, in the ratio of the focal 
distance of the objective to that of the eye-piece. But we 
have said that, seen through the objective, the image and 
the real object subtend the same angle. Hence the angu- 
lar magnifying power is equal to the focal distance of the 
objective, divided by that of the eye piece. If we simply 
turn the telescope end for end, the objective becomes the 
eye-piece and the latter the objective. The ratio is in- 



56 ASTRONOMY. 

verted, and the object is diminished in size in the same 
ratio that it is increased when viewed in the ordinary 
way. If we should form a telescope of two lenses of 
equal focal length, by placing them at double their focal 
distance, it would not magnify at all. 

The image formed by a convex lens, being upside 
down, and appearing in the same position when viewed 
with the eye-piece, it follows that the telescope, when 
constructed in the simplest manner, shows all objects in- 
verted, or upside down, and right side left. This is the 
case with all refracting telescopes made for astronomical 
uses. 

Light-gathering Power.— It is not merely by magnify- 
ing that the telescope assists the vision, but also by in- 
creasing the quantity of light which reaches the eye from 
the object at which we look. Indeed, should we view an 
object through an instrument which magnified, but did 
not increase the amount of light received by the eye, it is 
evident that the brilliancy would be diminished in propor- 
tion as the surface of the object was enlarged, since a con- 
stant amount of light would be spread over an increased 
surface ; and thus, unless the light were brilliant, the object 
might become so darkened as to be less plainly seen than 
with the naked eye. How the telescope increases the 
quantity of light will be seen by considering that when the 
unaided eye looks at any object, the retina can only re- 
ceive so many rays as fall upon the pupil of the eye. By 
the use of the telescope, it is evident that as many rays 
«an be brought to the retina as fall on the entire object- 
glass. The pupil of the human eye, in its normal state, 
has a diameter of about one fifth of an inch ; and by the 
use of the telescope it is virtually increased in surface in 
the ratio of the square of the diameter of the objective to 
the square of one fifth of an inch. Thus, with a two-inch 
aperture to our telescope, the number of rays collected is 
one hundred times as great as the number collected with 
the naked eye. 



POWER OF TELESCOPE. 57 

With a 5-incli object-glass, the ratio is 625 to 1 

" 10 " " " " " 2,500 to 1 

" 15 " " " " " 5,625 to 1 

" 20 " " " " " 10,000 to 1 

" 26 " " " " " 16,900 to 1 

When a minute object, like a star, is viewed, it is 
necessary that a certain number of rays should fall on the 
retina in order that the star may be visible at all. It is 
therefore plain that the use of the telescoj^e enables an 
observer to see much fainter stars than he could detect 
with the naked eye, and also to see faint objects much 
better than by unaided vision alone. Thus, with a 26- 
inch telescope we may see stars so minute that it would 
require many thousands to be visible to the unaided eye. 

An important remark is, however, to be made. here. 
Inspecting Fig. 20 we see that the cone of rays passing 
through the object-glass converges to a focus, then diverges 
at the same angle finally passing through the eye-piece. 
After this passage the rays emerge from the eye-piece 
parallel, as shown in Fig.- 22. It is evident that the 
diameter of this cylinder of parallel rays, or " emergent 
pencil," as it is called, is less than the diameter of the 
object-glass, in the same ratio that the focal length of the 
eye-piece is less than that of the object-glass. For the 
central ray I T is the common axis of two cones, A T and 
O T 0', having the same angle, and equal in length to 
the respective focal distances of the glasses. But this 
ratio is also the magnifying power. Hence the diameter 
of the emergent pencil of rays is found by dividing the 
diameter of the object-glass by the magnifying power. 
Now it is clear that if the magnifying power is so small 
that this emergent pencil is larger than the pupil of the 
eye, all the light which falls on the object-glass cannot 
enter the pupil. This will be the case whenever the 
magnifying power is less than five for every inch of 
aperture of the glass. If, for example, the observer should 



58 ASTRONOMY. 

look through a twelve-inch telescope with an eye-piece 
so large that the magnifying power was only 30, the 
emergent pencil would be two fifths of an inch in diam- 
eter, and only so much of the light could enter the pupil 
as fell on the central six inches of the object-glass. 
Practically, therefore, the observer would only be using a 
six-inch telescope, all the light which fell outside of the 
six-inch circle being lost. In order, therefore, that he 
may get the advantage of all his object-glass, he must use 
a magnifying power at least five times the diameter of his 
objective in inches. 

When the magnifying power is carried beyond this 
limit, the action of a telescope will depend partly on the 
nature of the object one is looking at. Viewing a star, 
the increase of power will give no increase of light, and 
therefore no increase in the apparent brightness of the 
star. If one is looking at an object having a sensible 
surface, as the moon, or a planet, the light coming 
from a given portion of the surface will be spread over a 
larger portion of the retina, as the magnifying power 
is increased. All magnifying must then be gained at 
the expense of the apparent illumination of the surface. 
Whether this loss of illumination is important or not will 
depend entirely on how much light is to spare. In a 
general way we may say that the moon and all the plan- 
ets nearer than Saturn are so brilliantly illuminated by 
the sun that the magnifying power can be carried many 
times above the limit without any loss in the distinctness 
of vision. 

The Telescope in Measurement. — A telescope is gen- 
erally thought of only as an instrument to assist the eye 
by its magnifying and light-gathering power in the man- 
ner we have described. But it has a very important 
additional function in astronomical measurements by en- 
abling the astronomer to point at a celestial object with a 
certainty and accuracy otherwise unattainable. This func- 
tion of the telescope was not recognized for more than 



USE OF TELESCOPE. 59 

half a century after its invention, and after a long and 
rather acrimonious contest between two schools of astron- 
omers. Until the middle of the seventeenth century, 
when an astronomer wished to determine the altitude of a 
celestial object, or to measure the angular distance be- 
tween two stars, he was obliged to point his quadrant or 
other measuring instrument at the object by means of 
i ' pinnules. ' ' These served the same purpose as the sights 
on a rifle. In using them, however, a difficulty arose. 
It was impossible for the observer to have distinct vision 
both of the object and of the pinnules at the same time, 
because when the eye was focused on either pinnule, or 
on the object, it was necessarily out of focus for the 
others. The only way to diminish this difficulty was to 
lengthen the arm on which the pinnules were fastened so 
that the latter should be as far apart as possible. Thus 
Tycho Brahe, before the year 1600, had measuring in- 
struments very much larger than any in use at the pres- 
ent time. But this plan only diminished the difficulty and 
could not entirely obviate it, because to be manageable 
the instrument must not be very large. 

About 1670 the English and French astronomers found 
that by simply inserting fine threads or wires exactly in 
the focus of the telescope, and then pointing it at the ob- 
ject, the image of that object formed in the focus could be 
made to coincide with the threads, so that the observer 
could see the two exactly superimposed upon each other. 
"When thus brought into coincidence, it was known that 
the point of the object on which the wires were set was in 
a straight line passing through the wires, and through the 
centre of the object-glass. So exactly could such a point- 
ing be made, that if the telescope did not magnify at all 
(the eye-piece and object-glass being of equal focal length), 
a very important advance would still be made in the ac- 
curacy of astronomical measurements. This line, passing 
centrally through the telescope, we call the line of col- 
limation of the telescope, A B in Fig. 20. If we have 



60 ASTRONOMY. 

any way of determining it we at once realize the idea ex- 
pressed in the opening chapter of this book, of a pencil ex- 
tended in a definite direction from the earth to the heav- 
ens. If the observer simply sets his telescope in a fixed 
position, looks through it and notices what stars pass along 
the threads in the eye-piece, he knows that those stars all 
lie in the line of colli mation of his telescope at that instant. 
By the diurnal motion, a pencil-mark, as it were, is thus 
being made in the heavens, the direction of which can be 
determined with far greater precision than by any meas- 
urements with the unaided eye. The direction of this line 
of collimation can be determined by methods which we 
need not now describe in detail. 

The Achromatic Telescope. — The simple form of tele- 
scope which we have described is rather a geometrical 
conception than an actual instrument. Only the earli- 
est instruments of this class were made with so few as two 
lenses. Galileo's telescope was not made in the form 
which we have described, for instead of two convex lenses 
having a common focus, the eye-piece was concave, and 
was placed at the proper distance inside of the focus of the 
objective. This form of instrument is still used in opera- 
glasses, but is objectionable in large instruments, owing to 
the smallness of the field of view. The use of two con- 
vex lenses was, we believe, first proposed by Kepler. 
Although telescopes of this simple form were wonderful 
instruments in their day, yet they would not now be re- 
garded as serving any of the purposes of such an instru- 
ment, owing to the absrrations with which a single lens is 
affected. We know that when ordinary light passes 
through a simple lens it is partially decomposed, the differ- 
ent rays coming to a focus at diff jrent distances. The 
focus for red rays is most distant from the object-glass, 
and that for violet rays the nearest to it, Thus arises 
the chromatic aberration of a lens. But this is not all. 
Even if the light is but of a single degree of refrangi- 
bility, if the surfaces of our lens are spherical, the rays 



ACHROMATIC OBJECT-GLASS. 



61 



which pass near the edge will come to a shorter focus 
than those which pass near the centre. Thus arises 
spherical aberration. This aberration might be avoided 
if lenses could be ground with a proper gradation of 
curvature from the centre to the circumference. Prac- 
tically, however, this is impossible ; the deviation from 
uniform sphericity, which an optician can produce, is too 
small to neutralize the defect. 

Of these two defects, the chromatic aberration is much 
the more serious ; and no way of avoiding it was known 
until the latter part of the last century. The fact had, 
indeed, been recognized by mathematicians and physicists, 
that if two glasses could be found having very different 
ratios of refractive to dispersive powers,* the defect could 
be cured by combining lenses made of these different 
kinds of glass. But this idea was not realized until the 
time of Dollond, an English optician who lived during 
the last century. This artist found that a concave lens of 
flint glass could be combined with a convex lens of crown of 
double the curvature in such a manner that the dispersive 
powers of the two lenses should neutralize each other, being 
equal and acting in opposite di- 
rections. But the crown glass 
having the greater refractive 
power, owing to its greater cur- 
vature, the rays would be brought 
to a focus without dispersion. 
Such is the construction of the 
achromatic objective. As now 
made, the outer or crown glass lens is double convex ; the 
inner or flint one is generally nearly plano-concave. 
Fig. 21 shows the section of such an objective as made 
by Alvan Clark & Sons, the inner curves of the crown 
and flint being nearly equal. 

*By the refractive power of a glass is meant its power of bending the 
rays out of their course, so as to bring them to a focus. By its disper- 
sive power is meant its power of separating the colors so as to form a 
spectrum, or to produce chromatic aberration. 




Fig. 21. — section of object 

GLASS. 



62 ASTRONOMY. 

A great advantage of the achromatic objective is that it 
may be made to correct the spherical as well as the chro- 
matic aberration. This is effected by giving the proper 
curvature to the various surfaces, and by making such 
slight deviations from perfect sphericity that rays passing 
through all parts of the glass shall come to the same focus. 

The Secondary Spectrum. — It is now known that the 
chromatic aberration of an objective cannot be perfectly 
corrected with any combination of glasses yet discovered. 
In the best telescopes the brightest rays of the spectrum, 
which are the yellow and green ones, are all brought to 
the same focus, but the red and blue ones reach a focus 
a little farther from the objective, and the violet ones a 
focus still farther. Hence, if we look at a bright star 
through a large telescope, it will be seen surrounded by a 
blue or violet light. If we push the eye-piece in a little 
the enlarged image of the star will be yellow in the centre 
and purple around the border. This separation of colors 
by a pair of lenses is called a secondary spectrum. 

Eye-Piece. — In the skeleton form of telescope before 
described the eye-piece as well as the objective was con- 
sidered as consisting of but a single lens. But with such 
an eye -piece vision is imperfect, except in the centre of 
the field, from the fact that the image does not throw 
rays in every direction, but only in straight lines away 
from the objective. Hence, the rays from near the edges 
of the focal image fall on or near the edge of the eye- 
piece, whence arises distortion of the image formed on 
the retina, and loss of light. To remedy this difficulty a 
lens is inserted at or very near the place where the focal 
image is formed, for the purpose of throwing the different 
pencils of rays which emanate from the several parts of 
the image toward the axis of the telescope, so that they 
shall all pass nearly through the centre of the eye lens pro- 
per. These two lenses are together called the eye-piece. 

There are some small differences of detail in the con- 
struction of eye-pieces, but the general principle is the 



THEORY OF OBJECT-GLASS. 



63 



same in all. The two recognized classes are the posi- 
tive and negative, the former being those in which the 
image is formed before the light reaches the held lens ; the 
negative those in which it is formed between the lenses. 

The figure shows the positive eye-piece drawn accurately to scale. 
O I is one of the converging pencils from the object-glass which 
forms one point (/) of the focal image la. This image is viewed 
by the field lens F of the eye-piece as a real object, and the shaded 
pencil between F and E shows the course of these rays after de- 
viation by F. If there were no eye-lens E an eye properly placed 
beyond F would see an image at T a '. The eye-lens E receives the 
pencil of rays, and deviates it to the observer's eye placed at such a 
point that the whole incident pencil will pass through the pupil 
and fall on the retina, and thus be effective. As we saw in the 




Fig 22. — section of a positive eye piece. 

figure of the refracting telescope, every point of the object produces 
a pencil similar to I, and the whole surfaces of the lenses F 
and E are covered with rays. All of these pencils passing through 
the pupil go to make up the retinal image. This image is referred 
by the mind to the distance of distinct vision (about ten inches), 
and the image A I" represents the dimension of the final image 

A F . 



a I 



relative to the image a I as formed by the objective and 

evidently the magnifying power of this particular eye-piece used 
in combination with this particular objective. 

More Exact Theory of the Objective.— For the benefit of the 
reader who wishes a more precise knov ledge of the optical princi- 
ples on which the action of the objective or other system of lenses 
depends, we present the following geometrical theory of the sub- 
ject. This theory is not rigidly exact, but is sufficiently so for all 
ordinary computations of the focal lengths and sizes of image in 
the usual combinations of lenses. 



64 



ASTRONOMY. 



Centres of Convergence and Divergence. —Suppose A B, Fig.. 
23, to be a lens or combination of lenses on which the light falls from 
the left hand and passes through to the right. Suppose rays parallel 
to RP to fall on every part of the first surface of the glass. After 
passing through it they are all supposed to converge nearly or ex- 
actly to the same point R'. Among all these rays there is one, and 
one only, the course of which, after emerging from the glass at Q, 
will be parallel to its original direction RP. Let R P Q R' be this 
central ray, which will be completely determined by the direction 
from which it comes. Next, let us take a ray coming from another 
direction as 8 P. Among all the rays parallel to S P, let us take 
that one which, after emerging from the glass at T, moves in a line 
parallel to its original direction. Continuing the process, let us 
suppose isolated rays coming from all parts of a distant object sub- 
ject to the single condition that the course of each, after passing 
through the glass or system of glasses, shall be parallel to its original 
course. These rays we may call central rays. They have this re- 
markable property, pointed out by Gauss : that they all converge- 




Fig. 23. 

toward a single point, P, in coming to the glass, and diverge from: 
another point, P', after passing through the last lens. These points 
were termed by Gauss " Hauptpunkte," or principal points. But 
they will probably be better understood if we call the first one the 
centre of convergence, and the second the centre of divergence. 
It must not be understood that the central rays necessarily pass 
through these centres. If one of them lies outside the first or last 
refracting surface, then the central rays must actually pass through 
it. But if they lie between the surfaces, they will be fixed by the 
continuation of the straight line in which the rays move, the latter 
being refracted out of their course by passing through the surface, 
and thus avoiding the points in question. If the lens or system of 
lenses be turned around, or if the light passes through them in an 
opposite direction, the centre of convergence in the first case be- 
comes the centre of divergence in the second, and vice versa. The 
necessity of this will be clearly seen by reflecting that a return ray 
of light will always keep on the course of the original ray in the 
opposite direction. 



THEORY OF OBJECT-GLASS. 65 

The figure represents a plano-convex lens with light falling on 
the convex side. In this case the centre of convergence will be 
on the convex surface, and that of divergence inside the glass 
about one third or two fifths of the way from the convex to the 
plane surface, the positions varying with the refractive index of the 
glass. In a double convex lens, both points will lie inside the glass, 
while if a glass is concave on one side and convex on the other, 
one of the points will be outside the glass on the concave side. It 
m ist be remembered that the positions of these centres of conver- 
gence and divergence depend solely on the form and size of the 
lenses and their refractive indices, and do not refer in any way to 
the distances of the objects whose images they form. 

The principal properties of a lens or objective, by which the size 
of images are determined, are as follows : Since the angle S' P R! 
made by the diverging rays is equal to R P S, made by the con- 
verging ones, it follows, that if a lens form the image of an object, 
the size of the image w T ill be to that of the object as their respec- 
tive distances from the centres of convergence and divergence. In 
other words, the object seen from the centre of convergence P will 
be of the same angular magnitude as the image seen from the 
centre of divergence P\ 

By conjugate foci of a lens or system of lenses we mean a pair of 
points such that if rays diverge from the one, they will converge to 
the other. Hence if an object is in one of a pair of such foci, the 
image will be formed in the other. 

By the refractive power of a lens or combination of lenses, we 
mean its influence in refracting parallel rays to a focus which we 
may measure by the reciprocal of its focal distance or 1 -~f. Thus, 
the power of a piece of plain glass is 0, because it cannot bring 
rays to a focus at all. The power of a convex lens is positive, while 
that of a concave lens is negative. In the latter case, it will be 
remembered by the student of optics that the virtual focus is on 
the same side of the lens from which the rays proceed. It is to 
be noted that when we speak of the focal distance of a lens, we 
mean the distance from the centre of divergence to the focus for 
parallel rays. In astronomical language this focus is called the 
stellar focus, being that for celestial objects, all of which we may 
regard as infinitely distant. If, now, we put 

p, the power of the lens ; 
f its stellar focal distance ; 

f, the distance of an object from the centre of convergence ; 
f', the distance of its image from the centre of divergence ; then 
the equation which determines/ will be 
1 1 1 

or, 

J— f + f ,\J -y_ f 

From these equations may be found the focal length, having the 
distance at which the image of an object is formed, or vice versa. 









66 



ASTRONOMY. 



% 2. REFLECTING TELESCOPES. 

As we have seen, the most essential part of a refracting 
telescope is the objective, which brings all the incident 
rajs from an object to one focus, forming there an image 
of that object. In reflecting telescopes (reflectors) the 
objective is a mirror of speculum metal or silvered glass 
ground to the shape of a paraboloid. The figure shows 
the action of such a mirroi on a bundle of parallel rajs, 
which, after impinging on it, are brought bj reflection to 
one focus F. The image formed at this focus maj be 
viewed with an eje-piece, as in the case of the refracting 
telescope. 

The eje- pieces used with such a mirror are of the kinds 
alreadj described. In the figure the eje-piece would 




Fig. 24. — concave mikrok forming an image. 

have to be placed to the right of the point F, and the 
observer's head would thus interfere with the incident 
light. Various devices have been proposed to remedj this 
inconvenience, of which we will describe the two most 
common. 

The Newtonian Telescope. — In this form the rajs of 
light reflected from the mirror are made to fall on a small 
plane mirror placed diagonally just before they reach the 
principal focus. The rays are thus reflected out laterally 
through an opening in the telescope tube, and are there 
brought to a focus, and the image formed at the point 
marked by a heavj white line in Fig. 25, instead of at 
the point inside the telescope marked by a dotted line. 



REFLECTING TELESCOPES. 



67 



This focal image is then examined by means of an or- 
dinary eye-piece, the head of the observer being outside 
of the telescope tube. 

This device is the invention of Sir Isaac Kewton. 



1 






5 






■* ; 



Fig. 25. 
kewtonian telescope. 




FTG. 26. 
CASSEGRAINIAN TELESCOPE. 






The Cassegrainian Telescope. — In this form a second- 
ary convex mirror is placed in the tube of the telescope 



68 ASTRONOMY 

about three fourths of the way from the large speculum 
to the focus. The rajs, after being reflected from the 
large speculum, fall on this mirror before reaching the 
focus, and are reflected back again to the speculum ; an 
opening is made in the centre of the latter to let the rays 
pass through. The position and curvature of the secondary 
mirror are adjusted so that the focus shall be formed just 
after passing through the opening in the speculum. 

In this telescope the observer stands behind or under 
the speculum, and, with the eye-piece, looks through the 
opening in the centre, in the direction of the object. 
This form of reflector is much more convenient in use 
than the Newtonian, in using which the observer has to 
be near the top of the tube. 

This form was devised by Cassegkain in 1672. 

The advantages of reflectors are found in their cheap- 
ness, and in the fact that, supposing the mirrors perfect in 
figure, all the rays of the spectrum are brought to one 
focus. Thus the reflector is suitable for spectroscopic or 
photographic researches without any change from its or- 
dinary form. This is not true of the refractor, since the 
rays by which we now photograph (the blue and violet 
rays) are, in that instrument, owing to the secondary 
spectrum, brought to a focus slightly different from that 
of the yellow and adjacent rays by means of which we 
see. 

Reflectors have been made as large as six feet in aper- 
ture, the greatest being that of Lord Rosse, but those 
which have been most successful have hardly ever been 
larger than two or three feet. The smallest satellite of 
Saturn (IRmas) was discovered by Sir William Herschel 
with a four-foot speculum, but ail the other satellites dis- 
covered by him were seen with mirrors of about eighteen 
inches in aperture. With these the vast majority of his 
faint nebulse were also discovered. 

The satellites of Neptune and Uranus were discovered 
by Lassell with a two-foot speculum, and much of the 



REFLECTING TELESCOPES. 69- 

work of Lord Rosse has been done with his three-foot 
mirror, instead of his celebrated six-foot one. 

From the time of Newton till quite recently it was 
usual to make the large mirror or objective out of specu- 
lum metal, a brilliant alloy liable to tarnish. When the 
mirror was once tarnished through exposure to the 
weather, it could be renewed only by a process of polish- 
ing almost equivalent to figuring and polishing the mirror 
anew. Consequently, in such a speculum, after the cor- 
rect form and polish were attained, there was great diffi- 
culty in preserving them. In recent years this difficulty 
has been largely overcome in two ways : first, by im- 
provements in the composition of the alloy, by which its 
liability to tarnish under exposure is greatly diminished, 
and, secondly, by a plan proposed by Foucault, whicli 
consists in making, once for all, a mirror of glass which 
will always retain its good figure, and depositing upon it a 
thin film of silver which may be removed and restored 
with little labor as often as it becomes tarnished. 

In this way, one important defect in the reflector has 
been avoided. Another great defect has been less success- 
fully treated. It is not a process of exceeding difficulty 
to give to the reflecting surface of either metal or glass 
the correct parabolic shape by which the incident rays are 
brought accurately to one focus. But to maintain this 
shape constantly when the mirror is mounted in a tube, 
and when this tube is directed in succession to various 
parts of the sky, is a mechanical problem of extreme diffi- 
culty. However the mirror may be supported, all the 
unsupported points tend by their weight to sag away from 
the proper position. When the mirror is pointed near 
the horizon, this effect of flexure is quite different from 
what it is when pointed near the zenith. 

As long as the mirror is small (not greater than eight to 
twelve inches in diameter), it is found easy to support it 
so that these variations in the strains of flexure have little 
practical effect. As we increase its diameter up to 48 or 






70 ASTRONOMY. 

72 inches, the effect of flexure rapidly increases, and 
special devices have to be nsed to counterbalance the 
injury done to the shape of the mirror. 

§ 3. CHRONOMETERS AND CLOCKS. 

In Chapter I. , § 8 , we described how the right ascen- 
sions of the heavenly bodies are measured by the times 
of their transits over the meridian, this quantity increas- 
ing by a minute of arc in four seconds of time. In order 
to determine it with all required accuracy, it is necessary 
that the time-pieces with which it is measured shall go 
with the greatest possible precision. There is no great 
difficulty in making astronomical measures to a second 
of arc, and a star, by its diurnal motion, passes over this 
space in one fifteenth of a second of time. It is there- 
fore desirable that the astronomical clock shall not vary 
from a uniform rate more than a few hundredths of a 
second in the course of a day. It is not, however, 
necessary that it should be perfectly correct ; it may go 
too fast or too slow without detracting from its char- 
acter for accuracy, if the intervals of time which it 
tells off — hours, minutes, or seconds — are always of ex- 
actly the same length, or, in other words, if it gains or 
loses exactly the same amount every hour and every day. 

The time-pieces used in astronomical observation are 
the chronometer and the clock. 

The chronometer is merely a very perfect time-piece 
with a balance-wheel so constructed that changes of tem- 
perature have the least possible effect upon the time of its 
oscillation. Such a balance is called a compensation bal- 
ance. 

The ordinary house clock goes faster in cold than in 
warm weather, because the pendulum rod shortens under 
the influence of cold. This effect is such that the clock 
will gain about one second a day for every fall of 3° Cent. 
(5°. 4 Fahr.) in the temperature, supposing the pendulum 



THE ASTRONOMICAL CLOCK. 



71 



rod to be of iron. Such changes of rate would be entirely 
inadmissible in a clock used for astronomical purposes. 
The astronomical clock is therefore provided with a com- 
pensation pendulum, by which the disturbing effects of 
changes of temperature are avoided. 

There are two forms now in use, the Harrison (grid- 
iron) and the mercurial. In the gridiron pendulum, the 
rod is composed in part of a number 
of parallel bars of steel and brass, 
so connected together that while the 
expansion of the steel bars produced 
by an increase of temperature tends 
to depress the hob of the pendulum, 
the greater expansion of the brass bars 
tends to raise it. When the total 
lengths of the steel and brass bars 
have been properly adjusted a nearly 
perfect compensation occurs, and the 
centre of oscillation remains at a con- 
stant distance from the point of sus- 
pension. The rate of the clock, so 
far as it depends on the length of the 
pendulum, will therefore be constant. 

In the mercurial pendulum the 
weight which forms the bob is a 
cylindric glass vessel nearly filled 
with mercury. With an increase of temperature the steel 
suspension rod lengthens, thus throwing the centre of 
oscillation away from the point of suspension ; at the 
same time the expanding mercury rises in the cylinder, 
and tends therefore to raise the centre of oscillation. 
When the length of the rod and the dimensions of the 
cylinder of mercury are properly proportioned, the centre 
of oscillation is kept at a constant distance from the point 
of suspension. Other methods of making this compensa- 
tion have been used, but these are the two in most gen- 
eral use for astronomical clocks. 




Fig. 27. —gridiron 
pendulum. 



72 ASTRONOMY. 

The correction of a chronometer (or clock) is the quantity of time 
(expressed in hours, minutes, seconds, and decimals of a second) 
which it is necessary to add algebraically to the indication of the 
hands, in order that the sum may be the correct time. Thus, if at 
sidereal h , May 18, at New York, a sidereal clock or chronometer 
indicates 23 h 58 m 20 s -7, its correction is + l m 39 s - 3; if atO' 1 (sidereal 
noon), of May 17, its correction was + l" 1 38 s -3, its daily rate or the 
change of its correction in a sidereal day is + I s - 0: in other words, 
this clock is losing 1" daily. 

For clock slow the sign of the correction is -j- ; 

u ,i jy lst u « u u a j s _. 

" " gaining" " " " rate is — ; 
" " losing " '' '• " " is -J- . 

A clock or chronometer may be well compensated for temperature, 
and yet its rate may be gaining or losing on the time it is intended 
to keep : it is not even necessary that the rate should be small (ex- 
cept that a small rate is practically convenient), provided only that 
it is constant. It is continually necessary to compute the clock cor- 
rection at a given time from its known correction at some other time, 
and its known rate. If for some definite instant we denote the time 
as shown by the clock (technically "the clock-face") by T, the true 
time by T and the clock correction by a T, we have 

T = T + a T 7 , and 
A T = T - T. 

Tn all observatories and at sea observations are made daily to de- 
termine A T. At the instant of the observation the time T is noted 
by the clock; from the data of the observation the time 7" is com- 
puted. If these agree, the clock is correct. If they differ, A /'is 
found from the above equations. 

If by observation we have found 

A To = the clock correction at a clock-time jT , 
A T = the clock correction at a clock-time T 7 , 
6T = the clock rate in a unit of time, 

we have 

AT = A T + dT(T— To) 

where T — To must be expressed in days, hours, etc., according as 
<5 T is the rate in one day, one hour, etc. 

When, therefore, the clock correction A To and rate 6T have been 
determined for a certain instant, T'o, we can deduce the true time 
from the clock-face T at any other instant by the equation T' = T 
+ A To + <57' (7'— 7'u). If the clock correction has been deter- 
mined at two different times, 7 7 and T to be A To and A T, the rate 
is inferred from the equation 

AT-ATo 

* T =-Tzrfr' 



THE ASTRONOMICAL CLOCK. 73 

These equations apply only so long as we can regard the rate as 
constant. As observations can be made only in clear weather, it is 
plain that during periods of overcast sky we must depend on these 
equations for our knowledge of T — i.e n the true time at a clock- 
time T. 

The intervals between the determination of the clock correction 
should be small, since even with the best clocks and chronometers 
too much dependence must not be placed upon the rate. The follow- 
ing example from Chauvenet's Astronomy will illustrate the practi- 
cal processes : 

lk Example. — At sidereal noon, May 5, the correction of a sidereal 
clock is— 16'" 47 s -0; at sidereal noon, May 12, it is — 10"' 13 s -50; 
what is the sidereal time on May 25, when the clock-face is ll u 13 m 
12 s • 6, supposing the rate to be uniform ? 

May 5, correction = — 16 m 47 s . 30 
" 12, kt = - 16 ,n 18 8 .50 



7 days' rate = -f- 33 s . 50 
6T = -r- 4 S .785 

Taking then as our starting-point T = May 12, 1 ', we have for the 
interval to T = May 25, ll h 13 m 12 s .6, T - to = 13 d ll h 13 m 12 8 .6 
= 13 d .467. Hence we have 

A To = - 16 m 13-.50 

8T(T- To) = -f l m 4\44 

AT = - 15 n ~9 8 T06 

T = ll h 13 m 12 8 .C0 

T' = 10"58 m 3 s . 54 

But in this example the rate is obtained for one true sidereal day, 
while the unit of the interval 13 d . 467 is a sidereal day as shown by 
the clock. The proper interval with which to compute the rate 
in this case is 13 d 10" 58 m 4 s . 13 = 13 d .457, with which we find 

aT = - 16'M3 s .50 

6TX 13.457 =-f l'» 4 s . 39 

♦ A T = — 15 m 9M1 

T = 1l h 13'" 12 8 .G0 



T = 10" 58'" 3 s . 49 



This repetition will be rendered unnecessary by always giving the 
rate in a unit of the clock. Thus, suppose that on June 3, at 4 h ll m 
12 s . 35 by the clock, we have found the correction -4- 2 m l(Kl4; 
and on June 4, at 14 h 17 m 49 9 .82 we have found the correction 4- 2 m 
19 8 .89; the rate in one hour of the clock will be 

_L Os 75 



74 ASTRONOMY. 

| 4. THE TRANSIT INSTRUMENT. 

The meridian transit instrument, or briefly the u tran- 
sit, " is used to observe the transits of the heavenly bodies, 



Q^ 




Fig. 28.— a transit instrument. 

and from the times of these transits as read from the 
clock to determine either the corrections of the clock or 
the right ascension of the observed body, as explained in 
Chapter I., § 8. 



THE TRANSIT INSTRUMENT. 



75 



It has two general forms, one (Fig. 28) for use in fixed 
observatories and one (Fig. 29) for use in the field. 

It consists essentially of a telescope T T (Fig. 28) 
mounted on an axis V V at right angles to it. 




Fig. 29. — pout able transit instrument. 

The ends of this axis terminate in accurately cylindrical 
steel pivots which rest in metallic bearings V V, in shape 
like the letter Y, and hence called the Ys. 



76 ASTRONOMY. 

These are fastened to two pillars of stone, brick, or 
iron. Two counterpoises W W are connected with the 
axis as in the plate, so as to take a large portion of the 
weight of the axis and telescope from the Ys, and thus to 
diminish the friction upon these and to render the rota- 
tion about V V more easy and regular. In the ordinary 
use of the transit, the line V V is placed accurately level 
and perpendicular to the meridian, or in the east and west 
line. To effect this " adjustment," there are two sets of 
adjusting screws, by which the ends of V V in the Ys may 
be moved either up and down or north and south. The 
plate gives the form of transit used in permanent observa- 
tories, and shows the observing chair C, the reversing car- 
riage /?, and the level L. The arms of the latter have 
Y's, which can be placed over the pivots V V. 

The line of collimation of the transit telescope is the 
line drawn through the centre of the objective perpendic- 
ular to the rotation axis V V. 

The reticle is a network of fine spider lines placed in 
the focus of the objective. 

In Fig. 30 the circle represents the field of view of a 
transit as seen through the eye-piece. The seven ver- 
tical lines, I, II, III, IV, Y, VI, 
VII, are seven fine spider lines 
tightly stretched across a metal plate 
or diaphragm, and so adjusted as to 
be perpendicular to the direction of 
a star's apparent diurnal motion. 
This metal plate can be moved right 
and left by fine screws. The hori- 
zontal wires, guide-wires, a and 5, 
mark the centre of the field. The 
field is illuminated at night by a lamp at the end of the 
axis which shines through the hollow interior of the lat- 
ter, and causes the field to appear bright. The wires are 
dark against a bright ground. The line of sight is a line 
joining the centre of the objective and the central one, IV, 
of the seven vertical wires. 




THE TRANSIT INSTRUMENT. 77 

The whole transit is in adjustment when, first, the axis 
V V is horizontal ; second, when it lies east and west ; 
and third, when the line of sight and the line of collima- 
tion coincide. When these conditions are fulfilled the 
line of sight intersects the celestial sphere in the meridian 
of the place, and when T T \& rotated about V Fthe line 
of sight marks out the meridian on the sphere. 

In practice the three adjustments are not exactly made, since it is 
impossible to effect them with mathematical precision. The errors 
of each of them are first made as small as is convenient, and are then 
determined and allowed for. 

To find the error of level, we place on the pivots a fine level (shown 
in position in the figure of the portable transit), and determine how 
much higher one pivot is than the other in terms of the divisions 
marked on the level tube. Such a level is shown in Fig. 4 of plate 
36, page 86. The value of one of these divisions in seconds of arc 
can be determined by knowing the length I of the whole level and 
the number n of divisions through which the bubble will run when 
one end is raised one hundredth of an inch. 

If I is the length of the level in inches or the radius of the circle 
in which either end of the level moves when it is raised, then as 
the radius of any circle is equal to 57°-296, 3437' -75 or 206, 264" -8, 
we have in this particular circle one inch == 206, 264" -8 -f- l\ 
01 inch = 206,264-8 -f- 100 I = a certain arc in seconds, say a". 
That is, n divisions = a", or one division d = a" ~ n. 

The error of collimation can be found by pointing the telescope 
at a distant mark whose image is brought to the middle wire. The 
telescope (with the axis) is then lifted bodily from the Ys and re- 
placed so that the axis V Fis reversed end for end. The telescope is 
again pointed to the distant mark. If this is still on the middle 
thread the line of sight and the line of collimation coincide. If not, 
the reticle must be moved bodily w T est or east until these conditions 
are fulfilled after repeated reversals. 

To find the error of azimuth or the departure of the direction of 
FFfromaneast and west line, we must observe the transits of 
two stars of different declinations 6 and d, and right ascensions a 
and a'. Suppose the clock to be running correctly — that is, with no 
rate — and the sidereal times of transit of the two stars over the mid- 
dle thread to be Q and B'. If — 0' = a — a', then the middle wire 
is in the meridian and the azimuth is zero. For if the azimuth 
was not zero, but the west end of the axis was too far south, for 
example, the line of sight would fall east of the meridian for a 
south star, and further and further east the further south the star 
was. Hence if the two stars have widely different declinations 6 
and 6', then the star furthest south would come proportionately 
sooner to the middle wire than the other, and — 0' would be 
different from a — a'. The amount of this difference gives a 



78 ASTRONOMY. 

means of deducing the deviation of A A from an east and west 
line. In a similar way the effect of a given error of level on the 
time of the transit of a star of declination 6 is found. 

Methods of Observing with the Transit Instrument.— 
We have so far assumed that the time of a star's transit 
over the middle thread was known, or could be noted. 
It is necessary to speak more in detail of how it is noted. 
When the telescope is pointed to any star the earth's 
diurnal motion will carry the image of the star slowly 
across the field of view of the telescope (which is kept 
fixed), as before explained. As it crosses each of the 
threads, the time at which it is exactly on the thread is 
noted from the clock, which must be near the transit. 

The mean of these times gives the time at which this 
star was on the middle thread, the threads being at equal 
intervals ; or on the " mean, thread," if, as is the case in 
practice, they are at unequal intervals. 

If it were possible for an astronomer to note the exact 
instant of the transit of a star over a thread, it is plain 
that one thread would be sufficient ; but, as all estima- 
tions of this time are, from the very nature of the case, 
but approximations, several threads are inserted in order 
that the accidental errors of estimations may be eliminated 
as far as possible. Five, or at most seven, threads are 
sufficient for this purpose. In the 
figure of the reticle of a transit instru- 
mentthe star (the planet Venus in this 
case) may enter on the right hand in the 
figure, and may be supposed to cross 
each of the wires, the time of its tran- 
sit over each of them, or over a suffi- 
cient number, being noted. The 
method of noting this time may be best 
understood by referring to the next figure. Suppose that 
the line in the middle of Fig. 32 is one of the transit- 
threads, and that the star is passing from the right hand 
of the figure toward the left ; if it is on this wire at an 




THE TRANSIT INSTRUMENT. 



79 




exact second by the clock (which is always near the ob- 
server, beating seconds audibly), this second must be writ- 
ten down as the time of the transit over this thread. As 
a rule, however, the transit cannot occur on the exact 
beat of the clock, but at the seventeenth second (for exam- 
ple) the star may be on the right of the wire, say at a ; 
while at the eighteenth second 
it will have passed this wire and 
may be at b. If the distance of 
a from the wire is six tenths of 
the distance ab, then the time 
of transit is to be recorded as — 
hours — ■ minutes (to be taken 
from the clock-face), and seven- 
teen and six tenths seconds; and in this way the transit 
over each wire is observed. This is the method of " eye- 
and-ear 1 ' observation, the basis of such work as we have 
described, and it is so called from the part which both the 
eye and the ear play in the appreciation of intervals of time. 
The ear catches the beat of the clock, the eye fixes the place 
of the star at a ; at the next beat of the clock, the eye fixes 
the star at b, and subdivides the space a b into tenths, at 
the same time appreciating the ratio which the distance 
from the thread to a bears to the distance a b. This is 
recorded as above. This method, which is still used in 
many observatories, was introduced by the celebrated 
Bradley, astronomer royal of England in 1T50, and per- 
fected by Maskelyne, his successor. A practiced observer 
can note the time within a tenth of a second in three cases 
out of four. 

There is yet another method now in common use, 
which it is necessary to understand. This is called the 
American or chronographic method, and consists, in the 
present practice, in the use of a sheet of a paper wound 
about and fastened to a horizontal cylindrical barrel, 
which is caused to revolve by machinery once in one min- 
ute of time. A pen of glass which will make a continu- 



•80 ASTRONOMY. 

ous line is allowed to rest on the paper, and to this pen a 
continuous motion of translation in the direction of the 
length of the cylinder is given. Now, if the pen is allow- 
ed to mark, it is evident that it will trace on the paper an 
endless spiral line. An electric current is caused to run 
through the observing clock, through a key which is held 
in the observer's hand and through an electro-magnet 
■connected with the pen. 

A simple device enables the clock every second to give 
a slight lateral motion to the pen, which lasts about a 
thirtieth of a second. Thus every second is automatically 
marked by the clock on the chronograph paper. The ob- 
server also has the power to make a signal by his key 
(easily distinguished from the clock-signal by its different 
length), which is likewise permanently registered on the 
sheet. In this way, after the chronograph is in motion, 
the observer has merely to notice the instant at which the 
star is on the thread, and to press the key at that moment. 
At any subsequent time, he must mark some hour, min- 
ute, and second, taken from the clock, on the sheet at its 
appropriate place, and the translation of the spaces on 
the sheet into times may be done at leisure. 

§ 5. GRADUATED CIRCLES. 

Nearly every datum in practical astronomy depends 
either directly or indirectly upon the measure of an angle. 
To make the necessary measures, it is customary to em- 
ploy what are called graduated or divided circles. These 
are made of metal, as light and yet as rigid as possible, 
and they have at their circumferences a narrow flat band 
of silver, gold, or platinum on which fine radial lines 
called " divisions' ' are cut by a " dividing engine' ■ at 
regular and equal intervals. These intervals may be 
of 10', 5', or 2', according to the size of the circle 
and the degree of accuracy desired. The narrow band 
is called the divided limb, and the circle is said to be di- 



THE VERNIER. 81 

vided to 10', 5', 2'. The separate divisions are numbered 
consecutively from 0° to 360° or from 0° to 90°, etc. The 
graduated circle has an axis at its centre, and to this may 
be attached the telescope by which to view the points 
whose angular distance is to be determined. 

To this centre is also attached an arm which revolves 
with it, and by its motion past a certain number of divi- 
sions on the circle, determines the angle through which the 
centre has been rotated. This arm is called the index 
arm, and it usually carries a vernier on its extremity, 
by means of which the spaces on 
the graduated circle are subdivided. 
The reading of the circle when the 
index arm is in any position is the 
number of degrees, minutes, and 
seconds which correspond to that po- 
sition ; when the index arm is in an- 
other position there is a different 
reading, and the differences of the two 
readings S' 2 — S\ S*—S% S 4 —S s are the angles through 
which the index arm has turned. 

The process of measuring the angle between the objects 
by means of a divided circle consists then of pointing the 
telescope at the first object and reading the position of the 
index arm, and then turning the telescope (the index 
arm turning with it) until it points at the second object, 
and again reading the position of the index arm. The 
difference of these readings is the angle sought. 

To facilitate the determination of the exact reading of 
the circle, we have to employ special devices, as the 
vernier and the reading microscope. 

The Vernier. — In Fig. 3-1, 31 i\ 7 is a portion of the 
divided limb of a graduated circle ; CD is the index arm 
which revolves with the telescope about the centre of the 
circle. The end a b of CD is also a part of a circle con- 
centric with 31 JV, and it is divided into n parts or divi- 
sions. The length of these n parts is so chosen that ir is 




Fig. 33. 



82 ASTRONOMY. 

the same as that of (n — 1) parts on the divided limb M N 
or the reverse. 

The first stroke a is the zero of the vernier, and the 
reading is always determined by the position of this zero 
or pointer. If this has revolved past exactly twenty di- 
visions of the circle, then the angle to be measured is 
20 X d, d being the value of one division on the limb 
(iV M) in arc. 




Fig. 34.— the vernier. 



Call the angular value of one division on the vernier d!\ 

n \ i 

(n — l)d = n-d\ord / = d, 2rt\did—d f =-d\ 

d — d' is called the least count of the vernier which is one 
n th part of a circle division. 

If the zero a does not fall exactly on a division on the 
circle, but is at some other point (as in the figure), for ex- 
ample between two divisions whose numbers are P and 
(P + 1), the whole reading of the circle in this position ie 
P x d + the fraction of a division from P to a. 

If the m th division of the vernier is in the prolongation 
of a division on the limb, then this fraction Pa is m 



THE MERIDIAN CIRCLE. 83 

(d — d f ) = --d. In the figure n = 10, and as the 4th 

division is almost exactly in coincidence, m = 4, so that 

the whole reading of the circle is P X d + r. • 6Z. If cHs 

10', for example, and if the division P is numbered 297° 
40', then this reading would be 297° 44', the least count 
being 1', and so in other cases. If the zero had started from 
the reading 280° 20', it must have moved past 17° 24', 
and this is the angle which has been measured. 



§ 6. THE MERIDIAN CIRCLE. 

The meridian circle is a combination of the transit in- 
strument with a graduated circle fastened to its axis and 
moving with it. The meridian circle made by E-epsold 
for the United States Naval Academy at Annapolis is 
shown in the figure. It has two circles, c c and c' c', finely 
divided on their sides. The graduation of each circle is 
viewed by four microscopes, two of which, P P, are 
shown in the cut. The microscopes are 90° apart. The 
cut shows also the hanging level L Z, by which the 
error of level of the axis A A is found. 

The instrument can be used as a transit to determine 
right ascensions, as before described. It can be also used 
to measure declinations in the following way. If the tele- 
scope is pointed to the nadir, a certain division of the cir- 
cles, as JV, is under the first microscope. If it is pointed 
to the pole, the reading will change by the angular distance 
between the nadir and the pole, or by 90° -f- </>, being 
the latitude of the place (supposed to be known). The 
polar reading P is thus known when the nadir reading 
JVis found. If the telescope is then pointed to various 
stars of unknown polar distances, p\ p'\p"\ etc., as they 
successively cross the meridian, and if the circle readings 
for these stars are P f , P" , P'" , etc. , it follows that 
p' = P'~P ; p" = P" - P ; y = P'» - />, etc. 



84 



ASTRONOMY. 




Pig. 35.— the meridian circle. 



THE MERIDIAN CIRCLE. 85 

To determine the readings P, JF", P", etc., we use the micro- 
scopes R, i?, etc. The observer, after having set the telescope so 
that one of the stars shall cross the field of view exactly at its cen- 
tre (which may be here marked by a single horizontal thread in 
the reticle), goes to each of the microscopes in succession and 
places his eye at A (see Fig. 1, page 8G). He sees in the field of the 
microscope the image of the divisions of the graduated scale (Fig. 2) 
formed at D (Fig. 1), the common focus of the lenses A and C. 
Just at that focus is placed a notched scale (Fig. 2) and two 
crossed spider lines. These lines are fixed to a sliding frame a a, 
which can be moved by turning the graduated head F. This head 
is divided usually into sixty parts, each of which is 1" of arc on 
the circle, one whole revolution of the head serving to move the 
sliding frame a a, and its crossed wires through 60" or 1' on the 
graduated circle. The notched scale is not movable, but serves to 
count the number of complete revolutions made by the screw, there 
being one notch for each revolution. The index i (Fig. 1) is fixed, 
and serves to count the number of parts of F which are carried past 
it by the revolution of this head. 

If on setting the crossed threads at the centre of the motion of 
F, and looking into the microscope, a division on the circle coin- 
cides with the cross, the reading of the circle P is the exact num- 
ber of degrees and minutes corresponding to that particular divi- 
sion on the divided circle. 

Usually, however, the cross has been apparently carried past one 
of the exact divisions of the circle by a certain quantity, which is 
now to be measured and added to the reading corresponding to 
this adjacent division. This measure can be made by turning the 
screw back say four revolutions (measured on the notched scale) 
plus 37*3 parts (measured by the index Tj. If the division of the 
circle in question was 179° 50', for example, the complete reading 
would be in this case 179° 50' + 4' 37''- 3 or 179° 54' 37". 3. Such 
a reading is made by each microscope, and the mean of the min- 
utes and seconds from all four taken as the circle reading. 

We now know how to obtain the readings of our circle when 
directed to any point. "We require some zero of reference, as 
the nadir reading (iV), the polar reading (P), the equator reading, 
•(Q), or the zenith reading (Z). Any one of these being known, the 
circle readings for any stars as P, P", P'', etc., can be turned into 
polar distances />', p", p"\ etc. 

The nadir reading (N) is the zero commonly employed. It can 
be determined by pointing the telescope vertically downward at 
a basin of mercury placed immediately beneath the instrument, and 
turning the whole instrument about the axis until the middle wire 
of the reticle seen directly exactly coincides with the image of 
this wire seen by reflection from the surface of the quicksilver. 
When this is the case, the telescope is vertical, as can be easily 
seen, and the nadir reading may be found from the circles. 
The meridian circle thus serves to determine both the right ascen- 
sion and declination of a given star at the same culmination. Zone 
observations are made with it by clamping the telescope in one 



86 



ASTRONOMY. 





Fi£.3, 




^ 




'Si 



Rfc 



r, s .4. 



fq 



Fig. 36. — reading microscope, micrometer and level 



THE EQUATORIAL. 87 

direction, and observing successively the stars which pass through 
its field of view. It is by this rapid method of observing that the 
largest catalogues of stars have been formed. 

§ 7. THE EQUATORIAL. 

To complete the enumeration and description of the 
principal instruments of astronomy, we require an account 
of the equatorial. This term, properly speaking, refers 
to a form of mounting, but it is commonly used to in- 
clude both mounting and telescope. In this class of 
instruments the object to be attained is in general the 
easy finding and following of any celestial object whose 
apparent place in the heavens is known by its right as- 
cension and declination. The equatorial mounting con- 
sists essentially of a pair of axes at right angles to each 
other. One of these S JV (the polar axis) is directed to- 
ward the elevated pole of the heavens, and it therefore 
makes an angle with the horizon equal to the latitude of 
the place (p. 23). This axis can be turned about its own 
axial line. On one extremity it carries another axis L D 
(the declination axis), which is fixed at right angles to it, 
but which can again be rotated about its axial line. 

To this last axis a telescope is attached, which may 
either be a reflector or a refractor. It is plain that such a 
telescope may be directed to any point of the heavens ; 
for we can rotate the declination axis until the telescope 
points to any given polar distance or declination. Then, 
keeping the telescope fixed in respect to the declination 
axis, we can rotate the whole instrument as one mass 
about the polar axis until the telescope points to any por- 
tion of the parallel of declination defined by the given 
right ascension or hour-angle. Fig. 37 is an equatorial of 
six-inch aperture which can be moved from place to place. 

If we point such a telescope to a star when it is rising 
(doing this by rotating the telescope first about its decli- 
nation axis, and then about the polar axis), and fix the 
telescope in this position, we can, by simply rotating the 



88 



ASTRONOMY, 




Fig. 37.— equatorial telescope pointed toward the pole. 



THE MICROMETER. 



89 



whole apparatus on the polar axis, cause the telescope to 
trace out on the celestial sphere the apparent diurnal path 
which this star will appear to follow from rising to set- 
ting. In such telescopes a driving-clock is so arranged 
that it can turn the telescope round the polar axis at the 
same rate at which the earth itself turns about its own axis 
of rotation, but in a contrary direction. Hence such a 
telescope once pointed at a star will continue to point at it 
as long as the driving-clock is in operation, thus enabling 
the astronomer to observe it at his leisure. 



s 


"0 

180 


%10° h 




w V 

N 


° \ 





Fig. 38.— measurement op position-angle. 



Every equatorial telescope intended for making exact measures 
has a filar micrometer, which is precisely the same in principle as 
the reading microscope in Fig. 2, page 86, except that its two wires 
are parallel. 

A figure of this instrument is given in Fig. 3, page 86. One of 
the wires is fixed and the other is movable by the screw. To 
measure the distance apart, of two objects A and B, wire 1 (the 
fixed wire) is placed on A and wire 2 (movable by the screw) is 
placed on B. The number of revolutions and parts of a revolution 
of the screw is noted, say 10 r -267 ; then wires I and 2 are placed 
in coincidence, and this zero-reading noted, say 5 r -143. The dis- 
tance A B is equal to 5 r - 124. Placing wires 1 and 2 a known num- 
ber of revolutions apart, we may observe the transits of a star in the 
equator over them ; and from the interval of time required for this 
star to move over say fifty revolutions, the value of one revolution 



90 ASTRONOMY. 

is known, and can always be used to turn distances measured in 
revolutions to distances in time or arc. 

By the filar micrometer we can determine the distance apart in 
seconds of arc of any two stars A and B. To completely fix the 
relative position of A and B, we require not only this distance, but 
also the angle which the line A B makes with some fixed direction 
in space. We assume as the fixed direction that of the meridian 
passing through A. Suppose in Fig. 38 A and B to be two 
stars visible in the field of the equatorial. The clock-work 
is detached, and by the diurnal motion of the earth the two 
stars will cross the field slowly in the direction of the parallel of 
declination passing through A, or in the direction of the arrow in 
the figure from E. to W., east to west. The filar micrometer is con- 
structed so that it can be rotated bodily about the axis of the tele- 
scope, and a graduated circle measures the amount of this rotation. 
The micrometer is then rotated until the star A will pass along 
one of its wires. This wire marks the direction of the parallel. 
The wire perpendicular to this is then in the meridian of the star. 

The position angle of B with respect to A is then the angle which 
A B makes with the meridian A N passing through A toward the 
north. It is zero when B is north of A, 90° when B is east, 180 
when B is south, and 270° when B is west of A. Knowing p, the 
position angle {N A Bin the figure), and s (A B) the distance of B, 
we can find the difference of right ascension (A a), and the differ- 
ence of declination (Ad) of B from A by the formulas, 

Ace = s sin p ; A<5 = % cos p. 

Conversely knowing Aa and Ad, we can deduce s and p from 
these formulas. The angle p is measured while the clock-work 
keeps the star A in the centre of the field. 

§ 8. THE ZENITH TELESCOPE. 

The accompanying figure gives a view of the zenith telescope in 
the form in which it is used by the United States Coast Survey. It 
consists of a vertical pillar which supports two Ys. In these 
rests the horizontal axis of the instrument which carries the tele- 
scope at one end, and a counterpoise at the other. The whole in- 
strument can revolve 180° in azimuth about this pillar. The tele- 
scope has a micrometer at its eye-end, and it also carries a divided 
circle provided with a fine level. A second level is provided, 
whose use is to make the rotation axis horizontal. The peculiar 
features of the zenith telescope are the divided circle and its at- 
tached level. The level is, as shown in the cut, in the plane of 
motion of the telescope (usually the plane of the meridian), and it 
can be independently rotated on the axis of the divided circle, and 
set by means of it to any angle with the optical axis of the telescope. 
The circle is divided from zero (0°) at its lowest point to 90° in 
each direction, and is firmly attached to the telescope tube, and 
moves with it. 

By setting the vernier or index-arm of the circle to any degree 
&nd minute as a, and clamping it there (the level moving with it) 



THE ZENITH TELESCOPE. 



91 




Fig. 39. — the zenith telescope. 



92 ASTRONOMY. 

and then rotating the telescope and the whole system about the 
horizontal axis until the bubble of the level is in the centre of the 
level-tube, the axis of the telescopes will be directed to the zenith 
distance a. The filar micrometer is so adjusted that a motion of its 
screw measures differences of zenith distance. The use of the ze- 
nith telescope is for determining the latitude by Talcott's 
method. The theory of this operation has been already given on 
page 48. A description of the actual process of observation will 
illustrate the excellences of this method. 

Two stars, A and B, are selected beforehand (from Star Cata- 
logues), which culminate, A south of the zenith of the place of ob- 
servation, B north of it. They are chosen at nearly equal zenith dis- 
tances £ A and £ B , and so that £ A - — £ B is less than the breadth of the 
field of view. Their right ascensions are also chosen so as to be about 
the same. The circle is then set to the mean zenith distance of the 
two stars, and the telescope is pointed so that the bubble is nearly in 
the middle of the level. Suppose the right ascension of A is the 
smaller, it will then culminate first. The telescope is then turned 
to the south. As A passes near the centre of the field its distance 
from the centre is measured by the micrometer. The level and 
micrometer are read, the whole instrument is revolved 180°, and 
star B is observed in the same way. 

By these operations we have determined the difference of the 
zenith distances of two stars whose declinations 6± and 6* are 
known. But o being the latitude. 

<p = <5 A + ^ and ? = c5 B — | B , whence 

The first term of this is known ; the second is measured ; so that 
each pair of stars so observed gives a value of the latitude which 
depends on the measure of a very small" -arc with the micrometer, 
and as this arc can be measured with great precision, the exactness 
of the determination of the latitude is equally great. - 

§ 9. THE SEXTANT. 

The sextant is a portable instrument by which the altitudes 
of celestial bodies or the angular distances between them may 
be measured. It is used chiefly by navigators for determining the 
latitude and the local time of the position of the ship. Knowing 
the local time, and comparing it with a chronometer regulated on 
Greenwich time, the longitude becomes known and the ship's place 
is fixed. 

It consists of the arc of a divided circle usually 60° in extent, 
whence the name. This arc is in fact divided into 120 equal parts, 
each marked as a degree, and these are again divided into smaller 
spaces, so that by means of the vernier at the end of the index-arm 
M S an arc of 10" (usually) may be read. 

The index-arm M S carries the index-glass 31, which is a silvered 
plane mirror set perpendicular to the plane of the divided arc. The 



THE SEXTANT. 



93 



horizon-glass m is also a plane mirror fixed perpendicular to the 
plane of the divided circle. 

This last glass is fixed in position, while the first revolves with 
the index-arm. The horizon-glass is divided into two parts, of 
which the lower one is silvered, the upper half being transparent. 
E is a telescope of low power pointed toward the horizon-glass. 
By it any object to which it is directed can be seen through the un- 
silvered half of the horizon-glass. Any other object in the same 
plane can be brought into the same field by rotating the index-arm 




Fig. 40.— the sextant. 

(and the index-glass with it), so that a beam of light from this 
second object shall strike the index-glass at the proper angle, there 
to be reflected to the horizon-glass, and again reflected down the 
telescope E. Thus the images of any two objects in the plane of 
tire sextant may be brought together in the telescope by viewing 
one directly, and the other by reflection. 

The principle upon which the sextant depends is the following, 
which is proved in optical works. The angle between the first and 
the last direction of a ray which has suffered tiro reflections in the soma 



94 ASTRONOMY. 

plane is equal to twice the angle which the two reflecting surfaces make 
with each other. 

In the figure S A is the ray incident upon A, and this ray is by 
reflection brought to the direction B E. The theorem declares 
that the angle B E S is equal to twice D C B, or twice the angle of 




Fig. 41. 

the mirrors, since B and D O are perpendicular to B and A. To 
measure the altitude of a star (or the sun) at sea, the sextant is held 
in the hand, and the telescope is pointed to the sea-horizon, which 
appears like a definite line. The index-arm is then moved until 
the reflected image of the sun or of the star coincides with the 




Fig. 42.— artificial horizon. 

image of the sea-horizon seen directly. When this occurs the time 
is to be noted from a chronometer. If a star is observed, the read- 
ing of the divided limb gives the altitude directly ; if it is the 
sun or moon which has been observed, the lower limb of these is 
brought to coincide with the horizon, and the altitude of the centre 



THE SEXTANT. 95 

is found by applying the semi-diameter as found in the Nautical 
Almanac to the observed altitude of the limb. 

The angular distance apart of a star and the moon can be meas- 
ured by pointing the telescope at the star, revolving the whole sex- 
tant about the sight-line of the telescope until the plane of the di- 
vided arc passes through both star and moon, and then by moving 
the index-arm until the reflected moon is just in contact with the 
star's image seen directly. 

On shore the horizon is broken up by buildings, trees, etc., and 
the observer is therefore obliged to have recourse to an artificial 
horizon, which consists usually of the reflecting surface of some 
liquid, as mercury, contained in a small vessel A, whose upper 
surface is necessarily parallel to the horizon DAG. A ray of light 
8 A, from a star at S, incident on the mercury at A, will be reflected 
in the direction A E, making the angle 8 A G = G A S' (A S' be- 
ing E A produced), and the reflected image of the star will appear 
to an eye at E as far below the horizon as the real star is above it. 
With a sextant whose index and horizon -glasses are at / and H, the 
angle 8 E 8' may be measured ; but SE8' = SAS' — A 8 E, 
and if ^4 E is exceedingly small as compared with A 8, as it is for 
all celestial bodies, the angle A 8 E may be neglected, and S E 8' 
will equal 8 A S', or double the altitude of the object : hence one 
half the reading of the instrument will give the apparent altitude. 



CHAPTER III. 

MOTION OF THE EARTH. 

§ 1. ANCIENT IDEAS OP THE PLANETS. 

It was observed by the ancients that while the great 
mass of the stars maintained their positions relatively to 
each other not only during each diurnal revolution, but 
month after month and } T ear after year, there were visi- 
ble to them seven heavenly bodies which changed their 
positions relatively to the stars and to each other. These 
they called planets or wandering stars. Still calling the 
apparent crystalline vault in which the stars seem to 
be set the celestial sphere, and imagining it as at rest, 
it was found that the seven planets performed a very 
slow revolution around the sphere from west to east, 
in periods ranging from one month in the case of the 
moon, to thirty years in that of Saturn. It was evident 
that these bodies could not be considered as set in the 
same solid sphere with the stars, because they could not 
then change their positions among the stars. Various 
ways of accounting for their motions were therefore pro- 
posed. One of the earliest conceptions is associated with 
the name of Pythagoras. He is said to have taught that 
each of the seven planets had its own sphere inside of and 
concentric with that of the fixed stars, and that these 
seven hollow spheres each performed its own revolution, 
independently of the others. This idea of a number of con- 
centric solid spheres was, however, apparently given up 



THE SOLAR SYSTEM. 



97 



without any one having taken the trouble to refute it by 
argument. Although at first sight plausible enough, a 
close examination would show it to be entirely inconsis- 
tent with the observed facts. The idea of the fixed stars 
being set in a solid sphere was, indeed, in seemingly 
perfect accord with their diurnal revolution as observed 
by the naked eye. But it was not so with the planets. 
The latter, after continued observation, were found to 
move sometimes backward and sometimes forward ; and 
it was quite evident that at certain periods they were 
nearer the earth than at other periods. These motions 
were entirely inconsistent with the theory that they were 
fixed in solid spheres. Still the old language continued in 
use — the word sphere meaning, not a solid body, but the 
space or region within which the planet moved. 

These several conceptions, as well as those which fol- 
lowed, were all steps toward the truth. The planets were 
rightly considered as bodies nearer to us than the fixed 
stars. It was also rightly judged that those which moved 
most slowly were the most distant, and thus their order of 
distance from the earth was correctly given, except in the 
case of Mercury and Venus. * 

We now know that these seven planets, together with 
the earth, and a number of other bodies which the tele- 
scope has made known to us, form a family or system by 
themselves, the dimensions of which, although inconceiv- 
ably greater than any which we have to deal with at the 
surface of the earth, are quite insignificant when com- 
pared with the distance w T hich separates us from the fixed 
stars. The sun being the great central body of this sys- 
tem, it is called the Solar System. It is to the motions of 
its several bodies and the consequences which flow from 
them that the attention of the reader is directed in the 
following chapters. We premise that there are now known 
to be eight large planets, of which the earth is the third 
in the order of distance from the sun, and that these 
bodies all perform a regular revolution around the sun. 



98 ASTRONOMY. 

Mercury, the nearest, performs its revolution in three 
months ; JVeptune, the farthest, in 164 years. 

First in importance to us, among the heavenly bodies 
which we see from the earth, stands the sun, the supporter 
of life and motion upon the earth. At first sight it might 
seem curious that the sun and seeming stars like Mars 
and Saturn should have been classified together as planets 
by the ancients, while the fixed stars were considered as 
forming another class. That the ancients were acute 
enough to do this tends to impress us with a favorable 
sense of the scientific character of their intellect. To any 
but the most careful theorists and observers, the star-like 
planets, if we may call them so, would never have seemed 
to belong in the same class with the sun, but rather in 
that of the stars ; especially when it was found that they 
were never visible at the same time with the sun. But 
before the times of which we have any historic record, 
there were men who saw that, in a motion from west to 
east among the fixed stars, these several bodies showed a 
common character, which was more essential in a theory 
of the universe than their immense differences of aspect 
and lustre, striking though these were. 

It must, however, be remembered that we no longer 
consider the sun as a planet. We have modified the an- 
cient system by making the sun and the earth change 
places, so that the latter is now regarded as one of the eight 
large planets, while the former has taken the place of the 
earth as the central body of the system. In consequence 
of the revolution of the planets round the sun, each of 
them seems to perform a corresponding circuit in the 
heavens around the celestial sphere, when viewed from 
any other planet or from the earth. 

§ 2. ANNUAL REVOLUTION OF THE EARTH. 

To an observer on the earth, the sun seems to perform an 
annual revolution among the stars, a fact which has been 
known from early ages. We now know that this motion 



MOTION OF THE EARTH. 



99 



is due to the annual revolution of the earth round the 
sun. It is to the nature and effects of this annual revolu- 
tion of the earth that the attention of the reader is now 
directed. Our first lesson is to show the relations between 
it and the corresponding apparent revolution of the sun, 
which is its counterpart. 

In Fig. 43, let S represent the sun, A B C D the orbit 
of the earth around it, and EF G II the sphere of the 




REVOLUTION OF THE EARTH. 



fixed stars. This sphere, being supposed infinitely dis- 
tant, must be considered as infinitely larger than the circle 
A B C D. Suppose now that 1, 2, 3, 4, 5, 6 are a 
number of consecutive positions of the earth. The line 
IS drawn from the sun to the earth in the first position is 
called the radius vector of the earth. Suppose this line 
extended infinitely so as to meet the celestial sphere in 
the point V. It is evident that to an observer on the 



100 ASTRONOMY. 

earth at 1 tlie sun will appear projected on the sphere 
in the direction of 1'. When the earth reaches 2, it 
will appear in the direction of 2', and so on. In other 
words, as the earth revolves around the sun, the latter 
will seem to perforin a revolution among the fixed stars, 
which are immensely more distant than itself. 

It is also evident that the point in which the earth would 
be projected, if viewed from the sun, is always exactly 
opposite that in which the sun appears as projected from 
the earth. Moreover, if the earth moves more rapidly in 
some points of its orbit than in others, it is evident Ithat 
the sun will also appear to move more rapidly among the 
stars, and that the two motions must always accurately 
correspond to each other. 

We now have the following definitions : 

The radius vector of the earth is the straight hne from 
the centre of the sun to the centre of the earth. 

As the earth describes its annual revolution around the 
sun, its radius vector describes a plane. This plane is 
called the plane of the ecliptic. 

If the plane of the ecliptic, being continued indefinitely 
in all directions, the great circle in which it cuts the ce^ 
lestial sphere is called the circle of the ecliptic, or simply 
the ecliptic. 

The axis of the ecliptic is a straight line passing through 
the centre of the sun at right angles to the plane of the 
ecliptic. 

The poles of the ecliptic are the two opposite points in 
which the axis of the ecliptic intersects the celestial 
sphere. 

Every point of the circle of the ecliptic is necessarily 
90° from each pole. 

.Effect of the sun^s annual motion upon the rising and 
setting of the stars. — It is evident from Fig. 43 that the 
sun appears to perform an annual revolution from west to 
east among the stars. Hence, if we watch any star for a 
few weeks, we shall find it to rise, cross the meridian, and 



THE SUN'S APPARENT PATH. 101 

set about 4 minutes earlier every day than it did the day 
before. 

Let us take, for example, the bright reddish star, Aide- 
baran, which, on a winter evening, we may see north- 
west of Orion. Near the end of February this star crosses 
the meridian about six o'clock in the evening, and sets 
about midnight. If we watch it night after night through 
the months of March and April, we shall find that it is far- 
ther and farther toward the west on each successive even- 
ing at the same hour. By the end of April we shall bare- 
ly be able to see it about the close of the evening twilight. 
At the end of May it will be so close to the sun as to be 
entirely invisible. This shows that during the months we 
have been watching it, the sun has been approaching the 
star' from the west. If in July we watch the eastern 
horizon in the early morning, we shall see this star rising 
before the sun. The sun has therefore passed by the 
star!, and is now east of it. At the end of November we 
will find it rising at sunset and setting at sunrise. The 
sun] is therefore directly opposite the star. During the 
winter months it approaches it again from the west, and 
passes it about the end of May, as before. Any other 
star south of the zenith shows a similar change, since the 
relative positions of the stars do not vary. 

§ 3. THE SUN'S APPARENT PATH. 

It is evident that if the apparent path of the sun lay in 
the equator, it would, during the entire year, rise exactly 
in the east and set in the west, and would always cross 
the meridian at the same altitude. The days would 
always be twelve hours long, for the same reason that a 
star in the equator is always twelve hours above the hori- 
zon and twelve hours below it. But we know that this 
is not the case, the sun being sometimes north of the 
equator and sometimes south of it, and therefore having 
a motion in declination. To understand this motion. 



102 



ASTRONOMY. 



suppose that on March 19th, 1879, the sun had been 
observed with a meridian circle and a sidereal clock at the 
moment of transit over the meridian of Washington. Its 
position would have been found to be this : 

Eight Ascension, 23 h 55 ra 23 s ; Declination, 0° 30' south. 

Had the observation been repeated on the 20th and 
following days, the results would have been : 

March 20, R. Ascen. 23 h 59 ra 2 s ; Dec. 0° 6' South. 
21, " h 2 m 40 s ; " 0° 17' North. 



22, 



Qh gm 19 s « QO 41 / ^ 0li ^ # 




Fig. 44.— the sun crossing the equator. 

If we lay these positions down on a chart, we shall find 
them to be as in Fig. 44, the centre of the sun being 
south of the equator in the first two positions, and north 
of it in the last two. Joining the successive positions by 
a line, we shall have a small portion of the apparent path 
of the sun on the celestial sphere, or, in other words, a 
small part of the ecliptic. 

It is clear from the observations and the figure that the 
sun crossed the equator between six and seven o'clock on 
the afternoon of March 20th, and therefore that the equa- 
tor and ecliptic intersect at the point where the sun was at 
that hour. This point is called the vernal equinox , the 



THE SUN'S APPARENT PATH. 



103 



first word indicating the season, while the second 
expresses the equality of the 
nights and days which occurs 
when the sun is on the equator. 
It will be remembered that this 
equinox is the point from which 
right ascensions are counted in 
the heavens in the same way 
that longitudes on the earth are 
counted from Greenwich or 
Washington. The sidereal ^ 
clock is therefore so set that P 
the hands shall read hours & 
minutes seconds at the ^ 
moment when the vernal equi- § 
nox crosses the meridian. § 

Continuing our observations §• 
of the sun's apparent course for > 
six months from March 20th £ 

fed 

till September 23d, we should g 
find it to be as in Fig. 45. It - * 
will be seen that Fig. 44 cor- £ 
responds to the right-hand end JJ 
of 45, but is on a much larger * 
scale. The sun, moving along | 
the great circle of the ecliptic, I 
will reach its greatest northern ' 
declination about June 21st. 
This point is indicated on the 
figure as 90° from the vernal 
equinox, and is called the sum- 
mer solstice. The sun's right 
ascension is then six hours, and 
its declination 23J° north. 

The course of the sun now 
inclines toward the south, and 
it again crosses the equator about September 22d at 




104 ASTRONOMY. 

a point diametrically opposite the vernal equinox. In 
virtue of the theorem of spherical trigonometry that all 
great circles intersect each other in two opposite points, 
the ecliptic and equator intersect at the two opposite equi- 
noxes. The equinox which the sun crosses on September 
22d is called the autumnal equinox. 

During the six months from September to March the 
sun's course is a counterpart of that from March to Sep- 
tember, except that it lies south of the equator. It at- 
tains its greatest south declination about December 22d, 
in right ascension 18 hours, and south declination 23^°. 
This point is called the winter solstice. It then begins to 
incline its course toward the north, reaching the vernal 
equinox again on March 20th, 1880. 
, The two equinoxes and the two solstices may be re- 
garded as the four cardinal points of the sun's apparent 
annual circuit around the heavens. Its passage through 
these points is determined by measuring its altitude or 
declination from day to day with a meridian circle. Since 
ip our latitude greater altitudes correspond to greater 
declinations, it follows that the summer solstice occurs on 
the day when the altitude of the sun is greatest, and the 
winter solstice on that when it is least. The mean of 
these altitudes is that of the equator, and may therefore 
be found by subtracting the latitude of the place from 
90°. The time when the sun reaches this altitude going 
north marks the vernal equinox, and that when it reaches 
it going south marks the autumnal equinox. 

These passages of the sun through the cardinal points 
have been the subjects of astronomical observation from 
the earliest ages on account of their relations to the change 
of the seasons. An ingenious method of finding the time 
when the sun reached the equinoxes was used by the as- 
tronomers of Alexandria about the beginning of our era. 
In the great Alexandrian Museum, a large ring or wheel 
was set up parallel to the plane of the equator — in other 
words, it was so fixed that a star at the pole would shine 



THE ZODIAC. 105 

perpendicularly on the wheel. Evidently its plane if 
extended must have passed through the east and west 
points of the horizon, while its inclination to the vertical 
was equal to the latitude of the place, which was not far 
from 30°. When the sun reached the equator going north 
or south, and shone upon this wheel, its lower edge would 
be exactly covered by the shadow of the upper edge ; 
whereas in any other position the sun would shine upon 
the lower inner edge. Thus the time at which the sun 
reached the equinox could be determined, at least to a 
fraction of a day. By the more exact methods of modern 
times, it can be determined within less than a minute. 

It will be seen that this method of determining the an- 
nual apparent course of the sun by its declination or alti- 
tude is entirely independent of its relation to the fixed 
.stars ; and it could be equally well applied if no stars 
were ever visible. There are, therefore, two entirely dis- 
tinct ways of finding when the sun or the earth has com- 
pleted its apparent circuit around the celestial sphere ; 
the one by the transit instrument and sidereal clock, which 
show when the sun returns to the same position among 
the stars, the other by the measurement of altitude, which 
shows when it returns to the same equinox. By the foiv 
mer method, already described, we conclude that it has 
completed an annual circuit when it returns to the same 
,star ; by the latter when it returns to the same equinox. 
These two methods will give slightly different results for 
the length of the year, for a reason to be hereafter 
described. 

The Zodiac and its Divisions. — The zodiac is a belt 
in the heavens, commonly considered as extending some S° 
-on each side of the ecliptic, and therefore about 16° wide. 
The planets known to the ancients are always seen within 
this belt. At a very early day the zodiac was mapped out 
into twelve signs known as the signs of t/n } zodiac, the 
names of which have been handed down to the present 
time. Each of these signs was supposed to be the seat of 



106 ASTRONOMY. 

a constellation after which it was called. Commencing 
at the vernal equinox, the first thirty degrees through 
which the sun passed, or the region among the stars in 
which it was found during the month following, was 
called the sign Aries. The next thirty degrees was called 
Taurus. The names of all the twelve signs in their 
proper order, with the approximate time of the sun's en- 
tering upon each, are as follow : 

Aries, the Ram, March 20. 

Taurus, the Bull, April 20. 

Gemini, the Twins, May 20. 

Cancer, the Crab, June 21. 

Leo, the Lion, July 22. 

Virgo, the Virgin, August 22. 

Libra, the Balance, September 22. 

Scorpius, the Scorpion, October 23. 

Sagittarius, the Archer, November 23. 

Capricornus, the Goat, December 21. 
Aquarius, the Water-bearer, January 20. 

Pisces, the Fishes, February 19. 

Each of these signs coincides roughly with a constella- 
tion in the heavens ; and thus there are twelve constella- 
tions called by the names of these signs, but the signs and 
the constellations no longer correspond. Although the sun 
now crosses the equator and enters the sign Aries on the 
20th of March, he does not reach the constellation Aries 
until nearly a month later. This arises from the preces- 
sion of the equinoxes, to be explained hereafter. 

§ 4. OBLIQUITY OP THE ECLIPTIC. 

We have already stated that when the sun is at the 
summer solstice, it is about 23£° north of the equator, 
and when at the winter solstice, about 23£° south. This 
shows that the ecliptic and equator make an angle 
of about 23J° with each other. This angle is called 



OBLIQUITY OF THE ECLIPTIC. 



107 



the obliquity of the ecliptic, and its determination is 
very simple. It is only necessary to find by repeated 
observation the sun's greatest north declination at the 
summer solstice, and its greatest south declination at 
the winter solstice. Either of these declinations, which 
must be equal if the observations are accurately made, 
will give the obliquity of the ecliptic. It has been con- 
tinually diminishing from the earliest ages at a rate of 
about half a second a year, or, more exactly , about forty- 
seven seconds in a century. This diminution is due to 
the gravitating forces of the planets, and will continue for 
several thousand years to come. It will not, however, go 
on indefinitely, but the obliquity will only oscillate be- 
tween comparatively narrow limits. 

The relation of the obliquity of the ecliptic to the sea- 
sons is quite obvious. When the sun is north of the equa- 
tor, it culminates at a higher altitude in the northern hem- 
isphere, and more than half of its apparent diurnal course 
is above the horizon, as explained in the chapter on the 
celestial sphere. Hence we have the heats of summer. 
In the southern hemisphere, of course, the case is re- 
versed : when the sun is in north declination, less than 
half of his diurnal course is above the horizon in that hem- 
isphere. Therefore this situation of the sun corresponds 
to summer in the northern hemisphere, and winter in the 
southern one. In exactly the same way, when the sun is 
far south of the equator, the days are shorter in the north- 
ern hemisphere and longer in the southern hemisphere. 
It is therefore winter in the former and summer in the 
latter. If the equator and the ecliptic coincided — that 
is, if the sun moved along the equator — there would 
be no such thing as a difference of seasons, because the 
sun would always rise exactly in the east and set exactly 
in the west, and always culminate at the same altitude. 
The days would always be twelve hours long the world 
over. This is the case with the planet Jupiter. 

In the preceding paragraphs, we have explained 



the 



108 ASTRONOMY. 

apparent annual circuit of the sun relative to the equator, 
and shown how the seasons depend upon this circuit. In 
order that the student may clearly grasp the entire subject, 
it is necessary to show the relation of these apparent move- 
ments to the aciual movement of the earth around the 
sun. 

To understand the relation of the equator to the eclip- 
tic, we must remember that the celestial pole and the 
celestial equator have really no reference whatever to the 
heavens, but depend solely on the direction of the earth's 
axis of rotation. The pole of the heavens is nothing 
more than that point of the celestial sphere toward which 
the earth's axis points. If the direction of this axis 
changes, the position of the celestial pole among the stars 
will change also ; though to an observer on the earth, 
unconscious of the change, it would seem as if the starry 
sphere moved while the pole remained at rest. Again, the 
celestial equator being merely the great circle in which the 
plane of the earth's equator, extended out to infinity in 
every direction, cuts the celestial sphere, any change in 
the direction of the pole of the earth necessarily changes 
the position of the equator among the stars. Xow the 
positions of the celestial pole and the celestial equator 
among the stars seem to remain unchanged throughout 
the year. (There is, indeed, a minute change, but it does 
not affect our present reasoning.) This shows that, as 
the earth revolves around the sun, its axis is constantly 
directed toward nearly the same point of the celestial 
sphere. 

§ 5. THE SEASONS. 

The conclusions to which we are thus led respecting 
the real revolution of the earth are shown -in Fig. 46. 
Here S represents the sun, with the orbit of the earth 
surrounding it, but viewed nearly edgeways so as to be 
much foreshortened. A B CD are the four cardinal 
positions of the earth which correspond to the cardinal 



IRE SEASOiVti. 109 

points of the apparent path of the sun already described. 
In each figure of the earth JVS is the axis, JV being its 
north and S its south pole. Since this axis points in the 




Fig. 46. — causes of the seasons. 

same direction relative to the stars during an entire year, 
it follows that the different lines N S are all parallel. 
Again, since the equator does not coincide with the ecliptic, 
these lines are not perpendicular to the ecliptic, but are 
inclined from this perpendicular by 23 £°. 

Now, consider the earth as at A / here it is seen that the 
sun shines more on the southern hemisphere than on the 
northern ; a region of 23-J- around the north pole is in 
darkness, while in the corresponding region around the 
south pole the sun shines all day. The five circles at right 
angles to the earth's axis are the parallels of latitude around 
which each region on the surface of the earth is carried by 
the diurnal rotation of the latter on its axis. It will be seen 
that in the northern hemisphere less than half of these are 
illuminated by the sun, and in the southern hemisphere 
more than half. This corresponds to our winter solstice. 

When the earth reaches B, its axis JVS is at right an- 
gles to the line drawn to the sun, so that the latter shines 
perpendicularly on the equator, the plane of which passes 
through it. The diurnal circles on the earth are one half 



110 ASTRONOMY. 

illuminated and one half in darkness. This position cor- 
responds to the vernal equinox. 

At C the case is exactly the reverse of that at A, the 
sun shining more on the northern hemisphere than on the 
southern one. North of the equator more than half the 
diurnal circles are in the illuminated hemisphere, and south 
of it less. Here then we have winter in the southern and 
summer in the northern hemisphere. The sun is above a 
region 23^° north of the equator, so that this position cor- 
responds to our summer solstice. 

At D the earth's axis is once more at right angles to a 
line drawn to the sun. The latter therefore shines upon 
the equator, and we have the autumnal equinox. 

In whatever position we suppose the earth, the line SJV 9 
continued indefinitely, meets the celestial sphere at its 
north pole, while the middle or equatorial circle of the 
earth, continued indefinitely in every direction, marks out 
the celestial equator in the heavens. At first sight it might 
seem that, owing to the motion of the earth through so 
vast a circuit, the positions of the celestial pole and equa- 
tor must change in consequence of this motion. We might 
say that, in reality, the pole of the earth describes a circle in 
the celestial sphere of the same size as the earth's orbit. 
But this sphere being infinitely distant, the circle thus de- 
scribed appears to us as a point, and thus the pole of the 
heavens seems to preserve its position among the stars 
through the whole course of the year. Again, we may 
suppose the equator to have a slight annual motion among 
the stars from the same cause. But for the same reason 
this motion is nothing when seen from the earth. On the 
other hand, the slightest change in the direction of the 
axis S JV will change the apparent position of the pole 
among the stars by an angle equal to that change of direc- 
tion. We may thus consider the position of the celestial 
pole as independent of the position of the earth in its 
orbit, and dependent entirely on the direction in which 
the axis of the earth points. 



CELESTIAL LATITUDE AND LONGITUDE. Ill 

If this axis were perpendicular to the plane of the eclip- 
tic, it is evident that the sun would always lie in the plane 
of the equator, and there would be no change of seasons 
except such slight ones as might result from the small 
differences in the distance of the earth at different seasons. 

§ 6. CELESTIAL LATITUDE AND LONGITUDE. 

Besides the circles of reference described in the first 
chapter, still another system is used in which the ecliptic 
is taken as the fundamental plane. Since the motion of 
the earth around the sun takes place, by definition, in the 
plane of the ecliptic, and the motions of the planets very 
near that plane, it is frequently more convenient to refer 
the positions of the planets to the plane of the ecliptic than 
to that of the equator. The co-ordinates of a heavenly 
body thus referred are called its celestial latitude and 
longitude. To show the relation of these co-ordinates to 
right ascension and declination, we give a figure showing 
both co-ordinates at the same time, as marked on the 
celestial sphere. This figure is supposed to be the celes- 
tial sphere, having the solar system in its centre. The 
direction P Q is that of the axis of the earth ; IJis the 
ecliptic, or the great circle in which the plane of the 
earth's orbit intersects the celestial sphere. The point in 
which these two circles cross is marked h , and is the ver- 
nal equinox. From this the right ascension and the longi- 
tude are counted, increasing from west toward east. 

The horizontal and vertical circles show how right ascen- 
sion and declination are counted in the manner described in 
Chapter I. As the right ascension is counted all the way 
around the equator from h to 24: h , so longitude is counted 
along the ecliptic from the point h , or the vernal equinox, 
toward J in degrees. The whole circuit measuring 360°, 
this distance will carry us all the way round. Thus if a 
body lies in the ecliptic, its longitude is simply the number 
of degrees from the vernal equinox to its position, meas- 
ured in the direction from /toward /(from west to east). 



112 



ASTRONOMY. 



If it is not in the ecliptic, but at, say, the point JB, we 
let fall a perpendicular B J from the body upon the 
ecliptic. The length of this perpendicular, measured in 
degrees, is called the latitude of the body, which may be 
north or south, while the distance of the foot of the per- 
pendicular from the vernal equinox is called its longitude. 
In astronomy it is usual to represent the positions of the 
hodies of the solar system, relatively to the sun, by their 
longitudes and latitudes, because in the ecliptic we have a 




Fig. 47. — circles of the sphere. 

plane more nearly fixed than that of the equator. On the 
other hand, it is more convenient to represent the position 
of all the heavenly bodies as seen from the earth by their 
right ascensions and declinations, because we cannot meas- 
ure their longitudes and latitudes directly, but can only 
observe right ascension and declination. If we wish to 
determine the longitude and latitude of a body as seen 
from the centre of the earth, we have to first find its right 
ascension and declination by observation, and then change 
these quantities to longitude and latitude by trigonometri- 
cal formulae. 



CHAPTER IV. 



THE PLANETARY MOTIONS. 



II. 



APPARENT AND REAL MOTIONS OF THE 
PLANETS. 



Definitions. — The solar system, as we now know it, com- 
prises so vast a number of bodies of various orders of mag- 
nitude and distance, and subjected to so many seemingly 
complex motions, that we must consider its parts sepa- 
rately. Our attention will therefore, in the present chap- 
ter, be particularly directed to the motions of the great 
planets, which we may consider as forming, in some sort, 
the fundamental bodies of the system. These bodies 
may, with respect to their apparent motions, be divided 
into three classes. 

Speaking, for the present, of the sun as a planet, the 
first class comprises the sun and moon. We have seen 
that if, upon a star chart, we mark down the positions of 
the sun day by day, they will all fall into a regular circle 
which marks out the ecliptic. The monthly course of the 
moon is found to be of the same nature, although its 
motion is by no means uniform in a month, yet it is 
always toward the east, and always along or very near a 
certain great circle. 

The second class comprises Venus and Mercury. The 
peculiarity exhibited by the apparent motion of these 
bodies is, that it is an oscillating one on each side of the 
sun. If we watch for the appearance of one of these 
planets after sunset from evening to evening, we shall find 



114 ASTRONOMY. 

it to appear above the western horizon. Night after night 
it will be farther and farther from the sun until it attains 
a certain maximum distance ; then it will appear to return 
to the sun again, and for a while to be lost in its rays. 
A few days later it will reappear to the west of the sun, 
and thereafter be visible in the eastern horizon before 
sunrise. In the case of Mercury, the time required for 
one complete oscillation back and forth is about four 
months ; and in the case of Venus more than a year and 
a half. 

The third class comprises Mars, Jupiter, and Saturn as 
well as a great number of planets not visible to the naked 
eye. The general or average motion of these planets is 
toward the east, a complete revolution in the celestial 
sphere being performed in times ranging from two years 
in the case of Mars to 164 years in that of Neptune. 
But, instead of moving uniformly forward, they seem to 
have a swinging motion ; first, they move forward or 
toward the east through a pretty long arc, then backward 
or westward through a short one, then forward through 
a longer one, etc. It is only by the excess of the longer 
arcs over the shorter ones that the circuit of the heavens 
is made. 

The general motion of the sun, moon, and planets 
among the stars being toward the east, the motion in this 
direction is called direct / whereas the occasional short 
motions toward the west are called retrograde. During 
the periods between direct and retrograde motion, the 
planets will for a short time appear stationary. 

The planets Venus and Mercury are said to be at great- 
est elongation when at their greatest angular distance from 
the sun. The elongation which occurs with the planet 
east of the sun, and therefore visible in the western hori- 
zon after sunset, is called the eastern elongation, the other 
the western one. 

A planet is said to be in conjunction with the sun when 
it is in the same direction, or when, as it seems to pass by 



ARRANGEMENT OF THE PLANETS. 



115 



the sun, it approaches nearest to it. It is said to be in 
opposition to the sun when exactly in the opposite direc- 
tion — rising when the sun sets, and vice versa. If, when 
a planet is in conjunction, it is between the earth and the 
sun, the conjunction is said to be an inferior one ; if be- 
yond the sun, it is said to be superior. 

Arrangements and Motions of the Planets. — We now 
know that the sun is the real centre of the solar system, 
and that the planets proper all revolve around it as the 
centre of motion. The order of the five innermost large 
planets, or the relative positions of their orbits, are shown 
in Fig. 48. These orbits are all nearly, but not exactly, 




Fig. 48. — orbits of the planets. 



in the same plane. The planets Mercury and Venus 
which, as seen from the earth, never appear to recede very 
far from the sun, are in reality those which revolve inside 



116 A8TB0N0MT. 

the orbit of the earth. The planets of the third class, 
which perform their circuits at all distances from the snn, 
are what we now call the superior planets, and are more 
distant from the sun than the earth is. Of these, the or- 
bits of Mars, Jupiter, and a swarm of telescopic planets 
are shown in the figure ; next outside of Jupiter comes 
Saturn, the farthest planet readily visible to the naked 
eye, and then Uranus and Neptune, telescopic planets. 
On the scale of Fig. 48 the orbit of Neptune would be 
more than two feet in diameter. Finally, the moon is a 
small planet revolving around the earth as its centre, and 
carried with the latter as it moves around the sun. 

Inferior planets are those whose orbits lie inside that 
of the earth, as Mercury and Venus. 

Superior planets are those whose orbits lie outside that 
of the earth, as Mars, Jupiter, Saturn, etc. 

The farther a planet is situated from the sun, the slower 
is its orbital motion. Therefore, as we go from the sun, 
the periods of revolution are longer, for the double reason 
that the planet has a larger orbit to describe and moves 
more slowly in its orbit. It is to this slower motion of the 
outer planets that the occasional apparent retrograde motion 
of the planets is due, as may be seen by studying Fig. 49. 
"We first remark that the apparent direction of a planet, 
as seen from the earth, is determined by the line joining 
the earth and planet. Supposing this line to be continued 
onward to infinity, so as to intersect the celestial sphere, 
the apparent motion of the planet will be defined by the 
motion of the point in which the line intersects the sphere. 
If this motion is toward the east, it will be direct; if 
toward the west, retrograde. 

Let us first take the case of an inferior planet. Sup- 
pose H I K L M N to be successive positions of the earth 
in its orbit, and A B C D E F to be corresponding posi- 
tions of Venus or Mercury. It must be remembered that 
when we speak of east and west in this connection, we do 
not mean an absolute direction in space, but a direction 



APPARENT MOTIONS OF THE PLANETS. 



11? 



around the sphere. In the figure we are supposed to look 
down upon the planetary orbits from the north, and a 
direction west is, then, that in which the hands of a watch 
move, while east is in the opposite direction. When the 
earth is at H the planet is seen at A . The line H A 
being supposed tangent to the orbit of the planet, it is 
evident from geometrical considerations that this is the 
greatest angle which the planet can ever make with the 
sun as seen from the earth. This, therefore, corresponds 
to the greatest eastern elongation. 







Fig. 49. 



When the earth has reached /the planet is at B, and is 
therefore near the direction IB. The line has turned in a 
direction opposite that of the hands of a watch, and cuts 
the celestial sphere at a point farther east than the line 
HA did. Hence the motion of the planet during this 
period has been direct ; but the direction of the sun hav- 
ing changed also in consequence of the advance of the 
earth, the angular distance between the sun and the planet 
is less than before. 

While the earth is passing from I to A", the planet 



118 ASTRONOMY. 

passes from B to C. The distance B C exceeds IK, be- 
cause the planet moves faster than the earth. The line 
joining the earth and planet, therefore, cuts the celestial 
sphere at a point farther west than it did before, and 
therefore the direction of the apparent motion is retro- 
grade. At C the planet is in inferior conjunction. The 
retrograde motion still continues until the earth reaches Z, 
and the planet D, when it becomes stationary. After- 
ward it is direct until the two bodies again come into the 
relative positions IB. 

Let us next suppose that the inner orbit A B CD EF 
represents that of the earth, and the outer one that of a 
superior planet, Mars for instance. We may consider 

Q P B to be the celestial sphere, only it should be infi- 
nitely distant. While the earth is moving from A to B the 
planet moves from H to I. This motion is direct, the di- 
rection Q P B being from west to east. While the earth 
is moving from B to D, the planet is moving from I to 

1 ; the former motion being the more rapid, the earth 
now passes by the planet as it were, and the line conjoin- 
ing them turns in the same direction as the hands of a 
watch. Therefore, during this time the planet seems to 
describe the arc P Q in the celestial sphere in the direction 
opposite to its actual orbital motion. The lines I D and 
M E are supposed to be parallel. The planet is then really 
stationary, even though as drawn it would seem to have a 
forward motion, owing to the distance of these two lines, 
yet, on the infinite sphere, this distance appears as a 
point. From the point M the motion is direct until the 
two bodies once more reach the relative positions B I 
When the planet is at K and the earth at O, the former is 
in opposition. Hence the retrograde motion of the supe- 
rior planets always takes place near opposition. 

Theory of Epicycles. — The ancient astronomers repre- 
sented this oscillating motion of the planets in a way which 
was in a certain sense correct. The only error they made 
was, in attributing the oscillation to a motion of the planet 



APPARENT MOTIONS OF THE PLANETS. 119 

instead, of a motion of the earth around the sun, which 
really causes it. But their theory was, notwithstanding, 
the means of leading Copernicus and others to the percep- 
tion of the true nature of the motion. We allude to the 
celebrated theory of epicycles, by which the planetary 
motions were always represented before the time of Coper- 
nicus. Complicated though these motions were, it was 
seen by the ancient astronomers that they could be repre- 
sented by a combination of two motions. First, a small 
circle or epicycle was supposed to move around the earth 
with a regular, though not uniform, forward motion, and 
then the planet was supposed to move around the circum- 
ference of this circle. The relation of this theory to the 
true one was this. The regular forward motion of the 
epicycle represents the real motion of the planet around 
the sun, while the motion of the planet around the cir- 
cumference of the epicycle is an apparent one arising 
from the revolution of the earth around the sun. To ex- 
plain this we must understand some of the laws of relative 
motion. 

It is familiarly known that if an observer in unconscious 
motion looks upon an object at rest, the object will ap- 
pear to him to move in a direction opposite that in 
which he moves. As a result of this law, if the observer 
is unconsciously describing a circle, an object at rest will 
appear to him to describe a circle of equal size. This is 
shown by the following figure. Let S represent the sun, 
and A B CD EF the orbit of the earth. Let us suppose 
the observer on the earth carried around in this orbit, but 
imagining himself at rest at S, the centre of motion. 
Suppose he keeps observing the direction and distance of 
the planet jP, which for the present we suppose to be at 
rest, since it is only the apparent motion that we shall 
have to consider. When the observer is at A he really 
sees the planet in a direction and distance A 7\ but 
imagining himself at S he thinks he sees the planet at 
the point a determined by drawing a line Sa parallel and 









120 



ASTRONOMY. 



equal to A P. As he passes from A to B the planet 
will seem to him to move in the opposite direction from 

a to b, the point b being deter- 
mined by drawing Sb equal and 
parallel to B P. As he recedes 
from the planet through the arc 
BCD, the planet seems to re- 
cede from him through bed; 
and while he moves from left to 
right through DE the planet 
seems to move from right to left 
through d e. Finally, as he ap- 
proaches the planet through the 
arc E FA the planet seems to 
approach him through efa, 
and when he returns to A the 
planet will appear at a, as in the 
beginning. Thus the planet, 
though really at rest, will seem 
to him to move over the circle 
abedef corresponding to that 
in which the observer himself is 
carried around the sun. 

The planet being really in motion, it is evident that 
the combined effect of the real motion of the planet and 
the apparent motion around the circle abe d ef will be 
represented by carrying the centre of this circle P along 
the true orbit of the planet. The motion of the earth 
being more rapid than that of an outer planet, it follows 
that the apparent motion of the planet through a b is more 
rapid than the real motion of P along the orbit. Hence 
in this part of the orbit the movement of the planet will be 
retrograde. In every other part it will be direct, because 
the progressive motion of P will at least overcome, some- 
times be added to, the apparent motion around the circle. 
In the ancient astronomy the apparent small circle 
abed ef was called the ejjicyele. 




Fig. 50. 



UNEQUAL MOTION OF THE PLANETS. 121 

In the case of the inner planets Mercury and Venus 
the relation of the epicycle to the true orbit is reversed. 
Here the epicyclic motion is that of the planet around 
its real orbit — that is, the true orbit of the planet around 
the sun was itself taken for the epicycle, while the 
forward motion was really due to the apparent revolu- 
tion of the sun produced by the annual motion of the 
earth. 

In the preceding descriptions of the planetary motions 
we have spoken of them all as circular. But it was found 
by IIipparchus * that none of the planetary motions were 
really uniform. Studying the motion of the sud in order 
to determine the length of the year, he observed the times 
of its passage through the equinoxes and solstices with all 
the accuracy which his instruments permitted. He found 
that it was several days longer in passing through one half 
of its course than through the other. This was apparently 
incompatible with the favorite theory of the ancients that 
all the celestial motions were circular and uniform. It 
was, however, accounted for by supposing that the earth 
was not in the centre of the circle around which the sun 
moved, but a little to one side. Thus arose the cele- 
brated theory of the eccentric. Careful observations of 
the jjlanets showed that they also had similar inequalities 
of motion. The centre of the epicycle around which the 
real planet was carried was found to move more rapidly 
in one part of the orbit, and more slowly in the opposite 
part. Thus the circles in which the planets were sup- 
posed to move were not truly centred upon the earth. 
They were therefore called eccentrics. 

This theory accounted in a rough way for the observed 
inequalities. It is evident that if the earth was supposed 
to be displaced toward one side of the orbit of the planet, 

* Hipparchus was one of the most celebrated astronomers of anti- 
quity, being frequently spoken of as the father of the science. He is 
supposed to have made most of his observations at Rhodes, and tlour 
ished about one hundred and fifty years before the Christian era. 






122 ASTRONOMY. 

the latter would seem to move more rapidly when nearest 
the earth than when farther from it. It was not until the 
time of Kepler that the eccentric was shown to be in- 
capable of accounting for the real motion ; and it is his 
discoveries which we are next to describe. 



§ 2. KEPLER'S LAWS OE PLANETARY MOTION. 

The direction of the sun, or its longitude, can be deter- 
mined from day to day by direct observation. If we 
could also observe its distance on each day, we should, by 
laying down the distances and directions on a large piece 
of paper, through a whole year, be able to trace the curve 
which the earth describes in its annual coarse, this course 
being, as already shown, the counterpart of the apparent 
one of the sun. A rough determination of the rela- 
tive distances of the sun at different times of the year may 
be made by measuring the sun's apparent angular diame- 
ter, because this diameter varies inversely as the distance 
of the object observed. Such measures would show that 
the diameter was at a maximum of 32' 36" on January 1st, 
and at a minimum of 31' 32" on July 1st of every year. 
The difference, 64", is, in round numbers, -^ the mean 
diameter — that is, the earth is nearer the sun on January 
1st than on July 1st by about ^ . We may consider it 
as -fo greater than the mean on the one date, and ^ less 
on the other. This is therefore the actual displacement 
of the sun from the centre of the earth's orbit. 

Again, observations of the apparent daily motion of 
the sun among the stars, corresponding to the real daily 
motion of the earth round the sun, show this motion to be 
least about July 1st, when it amounts to 57' 12" = 3432", 
and greatest about January 1st, when it amounts to 
61' 11" = 3671". The difference, 239", is, in round num- 
bers, -^j the mean motion, so that the range of variation 
is, in proportion to the mean, double what it is in the case 
of the distances. If the actual velocity of the earth in its 



KEPLER'S LAWS. 123 

orbit were uniform, the apparent angular motion round 
the sun would be inversely as its distance from the sun. 
Actually, however, the angular motion, as given above, is 
inversely as the square of the distance from the sun, be- 
cause (1 -f- -g^) 2 = 1 -(- tV very nearly. The actual ve- 
locity of the earth is therefore greater the nearer it is to 
the sun. 

On the ancient theory of the eccentric circle, as pro- 
pounded by Hipparchus, the actual motion of the earth 
was supposed to be uniform, and it was necessary to sup- 
pose the displacement of the sun (or, on the ancient theo- 
ry, of the earth) from the centre to be -^j its mean distance, 
in order to account for the observed changes in the motion 
in longitude. We now know that, in round numbers, one 
half the inequality of the apparent motion of the sun in 
longitude arises from the variations in the distance of the 
earth from it, and one half from the earth's actually mov- 
ing with a greater velocity as it comes nearer the sun. By 
attributing the whole inequality to a variation of distance, 
the ancient astronomers made the eccentricity of the or- 
bit — that is, the distance of the sun from the geometrical 
centre of the orbit (or, as they supposed, the distance of 
the earth from the centre of the sun's orbit) — twice as 
great as it really was. 

An immediate consequence of these facts of observa- 
tion is Kepler's second law of planetary motion, that the 
radii vector es drawn from the sun to a planet revolving 
round it, sweep over equal areas in equal times. Sup- 
pose, in Fig. 51, that S represents the position of the sun, 
and that the earth, or a planet, in a unit of time, say a 
day or a week, moves from P^ to P % . At another part 
of its orbit it moves from P to P t in the same time, 
and at a third part from P 4 to P b . Then the areas 
SP,P„ SPP„ SP,P b will all be equal. A little 
geometrical consideration will, in fact, make it clear that 
the areas of the triangles are equal when the angles at S 
are inversely as the square of the radii vectores, $ P, etc., 



124 ASTRONOMY. 

since the expression for the area of a triangle in which the 
angle at S is very small is % angle S X £ P 2 . * 




Fig. 51. — law of areas. 

In the time of Kepler the means of measuring the 
sun's angular diameter were so imperfect that the preced- 
ing method of determining the path of the earth around 
the sun could not be applied. It was by a study of the 
motions of the planet Mars, as observed by Tycho Brahe, 
that Kepler was led to his celebrated laws of planetary 
motion. He found that no possible motion of Mars in a 
truly circular orbit, however eccentric, would represent the 
observations. After long and laborious calculations, and 
the trial and rejection of a great number of hypotheses, 
he was led to the conclusion that the planet Mars moved 
in an ellipse, having the sun in one focus. As the analo- 
gies of nature led to the inference that all the planets, 
the earth included, moved in curves of the same class, 
and according to the same law, he was led to enunciate 
the first two of his celebrated laws of planetary motion, 
which were as follow : 

* More exactly if we consider the arc PPi as a straight line, the 
area of the triangle PPi S will be equal to | SPx SPi x sin angle S. 
But in considering only very small angles we may suppose SP= SPi 
and the sine of the angle S equal to the angle itself. This supposition 
will give the area mentioned above. 



KEPLER'S LAWS. 125 

I. Each planet m,oves around the sun in an ellipse, hav- 
ing the sun in one of its foci. 

II. The radius vector joining each planet with the 
sun, moves over equal areas in equal times. 

To these he afterward added another showing the rela- 
tion between the times of revolution of the separate 
planets. 

III. The square of the time of revolution of each 
planet is proportional to the cube of its mean distance 
from the sun. 

These three laws comprise a complete theory of plan- 
etary motion, so far as the main features of the motion are 
concerned. There are, indeed, small variations from 
these laws of Kepler, hut the laws are so nearly correct 
that they are always employed by astronomers as the basis 
of their theories. 

Mathematical Theory of the Elliptic Motion. — The 
laws of Kepler lead to problems of such mathematical 
elegance that we give a brief synopsis of the most impor- 
tant elements of the theory. A knowledge of the ele- 
ments of analytic geometry is necessary to understand it. 

Let us put : 

«, the semi-major axis of the ellipse in which the planet moves. 
In the figure, if G is the centre of the el- 
lipse, and S the focus in which the sun is 
situated, then a = A C = G tt. 

G S 
e, the eccentricity of the ellipse = — -. 

it, the longitude of the perihelion, rep- 
resented by the angle tt S E, E being the 
direction of the vernal equinox from, 
which longitudes are counted. 

n, the mean angular motion of the 
planet round the sun in a unit of time. 
The actual motion being variable, the 
mean motion is found by dividing the Fig. 52. 

circumference = 360° by the time of revolution. 

T, the time of revolution. 

r, the distance of the planet from the sun, or its radius vector, a 
variable quantity. 

I. The first remark we have to make is that the ellipticities of the 




126 ASTRONOMY. 

planetary orbits — that is, the proportions in which the orbits are flat- 
tened — is much less than their eccentricities. By the properties of 
the ellipse we have : 

S B = semi-major axis = a, 



B C= semi-minor axis = a Vl — e\ 
or, B C = a(l — ± e 2 ) nearly, when e is very small. 

The most eccentric of the orbits of the eight major planets is that 
of Mercury, for which e = 0.2. Hence for Mercury 

BC=a(l-T\) 

very nearly, so that flattening of the orbit is only about J^ or .02 
of the major axis. 

The next most eccentric orbit is that of Mars for which e = .093 ; 
B C = a (1 — .0043), so that the flattening of the orbit is only 
about ^ . 

We see from this that the hypothesis of the eccentric circle makes 
a very close approximation to the true form of the planetary orbits. 
It is only necessary to suppose the sun removed from the centre of 
the orbit by a quantity equal to the product of the eccentricity into 
the radius of the orbit to have a nearly true representation of the 
orbit and of the position of the sun. 

II. The least distance of the planet from the sun is 

3n = a(l-e), 
and the greatest distance is 

A S = a (I + e). 

III. The angular velocity of the planet around the sun at any 
point of the orbit, which we may call 8, is, by the second law of 
Kepler : 

r 2 

C being a constant to be determined. To determine it we remark 
that S is the angle through which the planet moves in a unit of 
time. If we suppose this unit to be very small, the quantity S r 2 is 
double the area of the very small triangle swept over by the radius 
Vector during such unit. This area is called the areolar velocity of 
the planet, and is a constant, by Kepler's second law. Therefore, 
in the last equation, O = S r 2 represents the double of the areolar 
velocity of the planet. When the planet completes an entire revo- 
lution, the radius vector has swept over the whole area of the 
ellipse which is tz a 2 V 1 — e 2 * The time required to do this be- 

* In this formula n represents the ratio of the circumference of the 
circle to its diameter. 






KEPLER'S LAWS 



127 



ing called T, the area swept over with the areolar velocity i G is 
also \G T. Therefore 



\ C T = it a? Vi — e 2 



C = 



The quantity 2 ^ here represents 360°, or the whole circumference, 
which, being divided by T, the time of describing it will give the 
mean angular velocity of the planet around the sun which we have 
called n. Therefore 



and 



C = a? n VI 



This value of C being substituted in the expression for 3, we have 



S= 



V\ 



IV. By Kepler's third law T* is proportioned to a? ; that is, 

— - is a constant for all the planets. The numerical value of this 

constant will depend upon the quantities which we adopt as the units 
of time and distance. If we take the year as the unit of time and 
the mean distance of the earth from the sun as that of distance, T 

and a for the earth will both be unity, and the ratio — r- will there- 

•" a 6 

fore be unity for all the planets. Therefore 
a 3 = T 2 ; a = T«. 

Therefore if we square the period of revolution of any planet in years, 
and extract the cube root of the square, tee shall have its mean distance 
from the sun in units of the earth's distance. 

It is thus that the mean distances of the planets are determined 
in practice, because, by a long series of observations, the times of 
revolution of the planets have been determined with very great pre- 
cision. 

V. To find the position of a planet we must know the epoch at 
which, it passed its perihelion, or some equivalent quantity. To 
find its position at any other time let r be the time w r hich has elapsed 
since passing the perihelion. Then, by the law of areas, if P be the 
position of the planet at this time we shall have 



Area of sector P Sir 



Area of whole ellipse 



(1). 



128 



ASTRONOMY. 



The times r and T being both given, the problem is reduced to 
that of cutting a given area of the ellipse by a line drawn from the 
focus to some point of its circumference to be found. This is 
known as Kepler' s problem, and may be solved by analytic geom- 




Fig. 53. 

etry as follows : Let A B be the major axis of the ellipse, Pthe 
position of the planet, and £ that of the focus in which the sun is 
situated. On A B as a diameter describe a circle, and through P 
draw the right line P* P D perpendicular to A B. 

The area of the elliptic sector SPB, over which the radius vector 
of the planet has swept since the planet passed the perihelion at B, 
is equal to the sector C P B minus the triangle C P S. Since an 
ellipse is formed from a circle by shortening all the ordinates in 
the same ratio (namely, the ratio of the minor axis h to the major 
axis a), it follows that the elliptic sector C P B may be formed 
from the circular sector C P B by shortening all the ordinates in 
the ratio of D P to D P*, or of a to I. Hence, 

Area CPB : area CP'B = b'.a. 

But area C P' B = angle P / C B x i a 2 , taking the unit radius 
as the unit of angular measure. Hence, putting u for the angle 
P* P we have 



Area CPB=- area CP' B = \alu 
a 



(2). 



Again, the area of the triangle CP S is equal to £ base C S x al- 
titude P D. Also P D = - P'D, and P> D = C P' sin u = a sin u. 
a 

Wherefore, 

PD=bs'mu (3). 






KEPLER 8 LAWS. 



129 



By the first principles of conic sections, C S, the base of the 
triangle, is equal to a e. Hence 

Area CP S = \ a h e sin u, (4) 

and, from (2) and (4), 

Area SP B = \ab {u — e sin u) . 

Substituting in equation (1) this value of the sector area, ana 
7r a b for the area of the ellipse, we have 

u — e sin u t 
2T ~ 7" 



e sin u = 2 7T 



From this equation the unknown angle u is to be found. The 
equation being a transcendental one, this cannot be done directly, 
but it may be rapidly done by successive approximation, or the 
value of u may be developed in an infinite series. 

Next we wish to express the position of the planet, which is given 
by its radius vector *S Pand the angle B IS P which this radius 
vector makes with the major axis of the orbit. Let us put 

r, the radius vector SP, 

f, the angle B S P, called the true anomaly. 

Then 

r sin/ = PD =: l> sin u (Equation 3), 

r cosf= S D = CD— C S = OP' cos u — ae = a (cos u — e), 

from which r and t /*can both be determined. By taking the square 
root of the sums of the squares, they give, by suitable reduction and 
putting o' 1 = a' (1 — <r), 

r = a (1 — e cos wi, 

and, by dividing the first by the second, 

b sin u 



tan/ = 



a (cos u — e) 



4 1 — e' 2 sin u 



cos it — e 



Putting, as before, n for the longitude of the perihelion, the true 
longitude of the planet in its orbit will be/ + t. 

VI. To find the position of the planet relatively to the ecliptic, 



130 • ASTRONOMY. 

the inclination of the orbit to the ecliptic has to be taken into ac- 
count. The orbits of the several large planets do not lie in the 
same plane, but are inclined to each other, and to the ecliptic, by 
various small angles. A table giving the values of these angles 
■will be given hereafter, from which it will be seen that the orbit of 
Mercury has the greatest inclination, amounting to 7°, and that of 
Uranus the least, being only 46'. The reduction of the position Oj! 
the planet to the ecliptic is a problem of spherical trigonometry, 
the solution of which need not be discussed here. 



CHAPTER V. 

UNIVERSAL GRAVITATION. 
§ 1. NEWTON'S LAWS OP MOTION. 

The establishment of the theory of universal gravitation 
furnishes one of the best examples of scientific method 
which is to be found. We shall describe its leading 
features, less for the purpose of making known to the 
reader the technical nature of the process than for illus- 
trating the true theory of scientific investigation, and 
showing that such investigation has for its object the dis- 
covery of what we may call generalized facts. The real 
test of progress is found in our constantly increased 
ability to foresee either the course of nature or the effects 
of any accidental or artificial combination of causes. So 
long as prediction is not possible, the desires of the inves- 
tigator remain unsatisfied. When certainty of prediction 
is once attained, and the laws on which the prediction is 
founded are stated in their simplest form, the work of 
science is complete. 

The whole process of scientific generalization consists in 
grouping facts, new and old, under such general laws that 
they are seen to be the result of those laws, combined with 
those relations in space and time which we may suppose to 
exist among the material objects investigated. It is essen- 
tial to such generalization that a single law shall suffice for 
grouping and predicting several distinct facts. A law 
invented simply to account for an isolated fact, however 



132 ASTRONOMY. 

general, cannot be regarded in science as a law of nature. 
It may, indeed, be true, but its truth cannot be proved 
until it is shown that several distinct facts can be accounted 
for by it better than by any other law. The reader will 
call to mind the old fable which represented the earth as 
supported on the back of a tortoise, but totally forgot that 
the support of the tortoise needed to be accounted for as 
much as that of the earth. 

To the pre-Xewtonian astronomers, the phenomena of the 
geometrical laws of planetary motion, which we have just 
described, formed a group of facts having no connection 
with any thing on the earth. The epicycles of Hipparchus 
and Ptolemy were a truly scientific conception, in that they 
explained the seemingly erratic motions of the planets by 
a single simple law. In the heliocentric theory of Coper- 
nicus this law was still further simplified by dispensing in 
great part with the epicycle, and replacing the latter by a 
motion of the earth around the sun, of the same nature 
with the motions of the planets. But Copernicus had no 
way of acco anting for, or even of describing with rigor- 
ous accuracy, the small deviations in the motions of the 
planets around the sun. In this respect he made no real 
advance upon the ideas of the ancients. 

Kepler, in his discoveries, made a great advance 
in representing the motions of all the planets by a 
single set of simple and easily understood geometrical 
laws. Had the planets followed his laws exactly, the 
theory of planetary motion would have been substantially 
complete. Still, further progress was desired for two 
reasons. In the first place, the laws of Kepler did not 
perfectly represent all the planetary motions. When ob- 
servations of the greatest accuracy were made, it was found 
that the planets deviated by small amounts from the ellipse 
of Kepler. Some small emendations to the motions com- 
puted on the elliptic theory were therefore necessary. 
Had this requirement been fulfilled, still another step 
would have been desirable — namely, that of connecting the 



LAWS OF MOTION. 



133 






motions of the planets with motion upon the earth, and 
reducing them to the same laws. 

Notwithstanding the great step which Kepler made in 
describing the celestial motions, he unveiled none of the 
great mystery in which they were enshrouded. This mys- 
tery was then, to all appearance, impenetrable, because 
not the slightest likeness could be perceived between the 
celestial motions and motions on the surface of the earth. 
The difficulty was recognized by the older philosophers in 
the division of motions into "forced" and "natural." 
The latter, they conceived, went on perpetually from the 
very nature of things, while the former always tended to 
cease. So when Kepler said that observation showed the 
law of planetary motion to be that around the circum- 
ference of an ellipse, as asserted in his law, he said all that 
it seemed possible to learn, supposing the statement per- 
fectly exact. And it was all that could be learned from the 
mere study of the planetary motions. In order to connect 
these motions with those on the earth, the next step was to 
study the laws of force and motion here around us. Sin- 
gular though it may appear, the ideas of the ancients on 
this subject were far more erroneous than their concep- 
tions of the motions of the planets. We might almost say 
that before the time of Galileo scarcely a single correct 
idea of the laws of motion was generally entertained by 
men of learning. There were, indeed, one or two who in 
this respect were far ahead of their age. Leonardo da 
Vinci, the celebrated painter, was noted in this respect. 
But the correct ideas entertained by him did not seem to 
make any headway in the world until the early part of 
the seventeenth century. Among those who, before the 
time of Newton, prepared the way for the theory in 
question, Galileo, Huyghens, and IIooke are entitled to 
especial mention. As, however, we cannot develop the 
history of this subject, we must pass at once to the gen- 
eral laws of motion laid down by Newton. These were 
three in number. 






134 ASTRONOMY. 

Law First : Every body preserves its state of rest or of 
uniform motion in a right line, unless it is compelled to 
change that state by forces impressed thereon. 

It was formerly supposed that a body acted on by no 
force tended to come to rest. Here lay one of the great- 
est difficulties which the predecessors of Newton found, 
in accounting for the motion of the planets. The idea 
that the sun in some way caused these motions was enter- 
tained from the earliest times. Even Ptolemy - had a 
vague idea of a force which was always directed toward 
the centre of the earth, or, which was to him the same 
thing, toward the centre of the universe, and which not 
only caused heavy bodies to fall, but bound the whole uni- 
verse together. .Kepler, again, distinctly affirms the ex- 
istence of a gravitating force by which the sun acts on the 
planets ; but he supposed that the sun must also exercise 
an impulsive forward force to keep the planets in motion. 
The reason of this incorrect idea was, of course, that all 
bodies in motion on the surface of the earth had practically 
come to rest. But what was not clearly seen before the 
time of Newton, or at least before Galileo, was, that this 
arose from the inevitable resisting forces which act upon 
all moving bodies around us. 

Law Second : The alteration of motion is ever propor- 
tional to the moving force impressed, and is made in the 
direction of the right line in which that force acts. 

The first law might be considered as a particular case of 
this second one arising when the force is supposed to van- 
ish. The accuracy of both laws can be proved only by 
very carefully conducted experiments. They are now 
considered as mathematically proved. 

Law Third : To every action there is always opposed an 
equal reaction * or the mutual actions of two bodies %ipon 
each other are always equal, and in opposite directions. 

That is, if a body A acts in any way upon a body B, 
B will exert a force exactly equal on A in the opposite 
direction. 



GRAVITATION OF THE PLANETS. 



135 



These laws once established, it became possible to calcu- 
late the motion of any body or system of bodies when once 
the forces which act on them were known, and, vice versa, 
to define what forces were requisite to produce any given 
motion. The question which presented itself to the mind 
of Newton and his contemporaries was this : Under what 
law of force will planets move round the sun in accord- 
ance with Kepler's laws f 

The laws of central forces had been discovered by Huy- 
ghens some time before Newton commenced his re- 
searches, and there was one result of them which, taken in 
connection with Kepler's third law of motion, was so 
obvious that no mathematician could have had much diffi- 
culty in perceiving it. Supposing a body to move around 
in a circle, and putting R the radius of the circle, T the 
period of revolution, Huyghens showed that the centrifugal 
force of the body, or, which is the same thing, the attract- 
ive force toward the centre which would keep it in the 

circle, was proportional to 7 ^. But by Kepler's third 
law T* is proportional to E z . Therefore this centripetal 
force is proportional to -^, that is, to -y^. Thus it fol- 
lowed immediately from Kepler's third law, that the 
central force which would keep the planets in their or- 
bits was inversely as the square of the distance from the 
sun, supposing each orbit to be circular. The first law of 
motion once completely understood, it was evident that 
the planet needed no force impelling it forward to keep 
up its motion, but that, once started, it would keep on 
forever. 

The next step was to solve the problem, what law of 
force will make a planet describe an ellipse around the 
sun, having the latter in one of its foci ? Or, supposing 
a planet to move round the sun, the latter attracting it 
with a force inversely as the square of the distance ; what 
will be the form of the orbit of the ..planet if it is not cir- 



136 ASTRONOMY. 

cular ? A solution of either of these problems was beyond 
the power of mathematicians before the time of Newton ; 
and it thus remained uncertain whether the planets mov- 
ing under the influence of the sun's gravitation would or 
w r ould not describe ellipses. Unable, at first, to reach a 
satisfactory solution, Newton attacked the problem in 
another direction, starting from the gravitation, not of 
the sun, but of the earth, as explained in the following 
section. 



§ 2. GRAVITATION IN THE HEAVENS. 

The reader is probably familiar with the story of New- 
ton and the falling apple. Although it has no authorita- 
tive foundation, it is strikingly illustrative of the method 
by which Newton first reached a solution of the problem. 
The course of reasoning by which he ascended from grav- 
itation on the earth to the celestial motions was as follows : 
We see that there is a force acting all over the earth by 
which all bodies are drawn toward its centre. This force 
is familiar to every one from his infancy, and is properly 
called gravitation. It extends without sensible diminution 
to the tops not only of the highest buildings, but of the 
highest mountains. How much higher does it extend ? 
Why should it not extend to the moon ? If it does, the 
moon would tend to drop toward the earth, just as a stone 
thrown from the hand drops. As the moon moves round 
the earth in her monthly course, there must be some force 
drawing her toward the earth ; else, by the first law of 
motion, she would fly entirely away in a straight line. Why 
should not the force which makes the apple fall be the 
same force which keeps her in her orbit ? To answer this 
question, it was not only necessary to calculate the intensity 
of the force which would keep the moon herself in her 
orbit, but to compare it with the intensity of gravity at the 
earth's surface. It had long been known that the distance 
of the moon was about sixty radii of the earth. If this 



GRAVITATION OF THE PLANETS. 



13? 



force diminished as the inverse square of the distance, 
then, at the moon, it would be only -3 gVo as g reat as at 
the surface of the earth. On the earth a body falls six- 
teen feet in a second. If, then, the theory of gravitation 
were correct, the moon ought to fall toward the earth 
tbVtt °^ ^ ms amoun t> or about A- f a n inch in a second. 
The moon being in motion, if we imagine it moving in a 
straight line at the beginning of any second, it ought to 
be drawn away from that line T ^ of an inch at the end of 
the second. When the calculation was made with the 
correct distance of the moon, it was found to agree ex- 
actly with this result of theory. Thus it was shown that 
the force which holds the moon in her orbit is the same 
which makes the stone fall, only diminished as the inverse 
square of the distance from the centre of the earth.* 

As it appeared that the central forces,- both toward the 
sun and toward the earth, varied inversely as the squares 
of the distances, Newton proceeded to attack the mathe- 
matical problems involved in a more systematic way than 
any of his predecessors had done. Kepler's second law 
showed that the line drawn from the planet to the sun 
will describe equal areas in equal times. Newton showed 
that this could not be true, unless the force which held 
the planet was directed toward the sun. We have already 
stated that the third law showed that the force was in- 
versely as the square of the distance, and thus agreed ex- 
actly with the theory of gravitation. It only remained to 



* It is a remarkable fact in the history of science that Newton 
would have reached this result twenty years sooner than he did, had 
he not been misled by adopting an erroneous value of the earth's diame- 
ter. His first attempt to compute the earth's gravitation at the distance 
of the moon was made in 1665, when he was only twenty-three years of 
age. At that time he supposed that a degree on the earth's surface was 
sixty statute miles, and was in consequence led to erroneous results In- 
supposing the earth to be smaller and the moon nearer than they really 
were. He therefore did not make public his ideas ; but twenty years 
later he learned from the measures of Picard in France what the true 
diameter of the earth was, when he repeated his calculation with 
entire success. 



138 ASTRONOMY. 

consider the results of the first law, that of the elliptic 
motion. After long and laborious efforts, Xewton was 
enabled to demonstrate rigorously that this law also re- 
sulted from the law of the inverse square, and could result 
from no other. Thus all mystery disappeared from the 
celestial motions ; and planets were shown to be simply 
heavy bodies moving according to the same laws that were 
acting here around us, only under very different circum- 
stances. All three of Kepler's laws were embraced in 
the single law of gravitation toward the sun. The sun 
attracts the planets as the earth attracts bodies here 
around us. 

Mutual Action of the Planets. — It remained to extend 
and prove the theory by considering the attractions of the 
planets themselves. By Newton's third law of motion, 
each planet must attract the sun with a force equal to that 
which the sun exerts upon the planet. The moon also 
must attract the earth as much as the earth attracts the 
moon. Such being the case, it must be highly probable 
that the planets attract each other. If so, Kepler's laws 
can only be an approximation to the truth. The sun, 
being immensely more massive than any of the planets, 
overpowers their attraction upon each other, and makes 
the law of elliptic motion very nearly true. But still the 
comparatively small attraction of the planets must cause 
some deviations. Now, deviations from the pure elliptic 
motion were known to exist in the case of several of the 
planets, notably in that of the moon, which, if gravitation 
were universal, must move under the influence of the com- 
bined attraction of the earth and of the sun. Newton, 
therefore, attacked the complicated problem of the deter- 
mination of the motion of the moon under the combined 
action of these two forces. He showed in a general way 
that its deviations would be of the same nature as those 
shown by observation. But the complete solution of the 
problem, which required the answer to be expressed in 
numbers, was beyond his power. 









ATTRACTION OF GRAVITATION. 



139 



Gravitation Besides in each Particle of Matter. — Still 
another question arose. Were these mutually attractive 
forces resident in the centres of the several bodies attracted, 
or in each particle of the matter composing them ? New- 
ton showed that the latter must be the case, because the 
smallest bodies, as well as the largest, tended to fall 
toward the earth, thus showing an equal gravitation in 
every separate part. The question then arose : what 
would be the action of the earth upon a body if the 
body was attracted — not toward the centre of the earth 
alone, but toward every particle of matter in the earth ? 
It was shown by a quite simple mathematical demonstra- 
tion that if a planet were on the surface of the earth or 
outside of it, it would be attracted with the same force as 
if the whole mass of the earth were concentrated in its 
centre. Putting together the various results thus arrived 
at, Newton was able to formulate his great law of uni- 
versal gravitation in these comprehensive words : " Every 
f article of matter in the universe attracts every other 
particle with a force directly as the masses of the two 
particles, and inversely as the square of the distance 
which separates them. ' ' 

To show the nature of the attractive forces anions 
these various particles, let us represent by m and m! the 
masses of two attracting bodies. We may conceive the 
body m to be composed of m particles, and the other 
body to be composed of m' particles. Let us conceive that 
each particle of the one body attracts each particle of the 

other with a force — a . Then every particle of m will be 

attracted by each of the m! particles of the other, and 

therefore the total attractive force on each of these m par- 

m! 
tides will be — . Each of the m particles being equally 

subject to this attraction, the total attractive force between 



the two bodies will be 



m m 



When a 



given force acts 



140 ASTRONOMY. 

upon a body, it will produce less motion the larger the 
body is, the accelerating force being proportional to the 
total attracting force divided by the mass of the body 
moved. Therefore the accelerating force which acts on the 
body m', and which determines the amount of motion, will 

be — 2 - ; and conversely the accelerating force acting on the 

771 

body 77i will be represented by the fraction — r . 



§ 3. PROBLEMS OF GRAVITATION. 

The problem solved by Newton, considered in its great- 
est generality, was this : Two bodies of which the masses 
are given are projected into space, in certain directions, and 
with certain velocities. What will be their motion under 
the influence of their mutual gravitation ? If their rela- 
tive motion does not exceed a certain definite amount, they 
will each revolve around their common centre of gravity 
in an ellipse, as in the case of planetary motions. If, how- 
ever, the relative velocity exceeds a certain limit, the two 
bodies will separate forever, each describing around the 
common centre of gravity a curve having infinite branches. 
These curves are found to be parabolas in the case where 
the velocity is exactly at the limit, and hyperbolas when 
the velocity exceeds it. Whatever curves may be de- 
scribed, the common centre of gravity of the two bodies 
will be in the focus of the curve. Thus, when restricted 
to two bodies, the problem admits of a perfectly rigorous 
mathematical solution. 

Having succeeded in solving the problem of planetary 
motion for the case of two bodies, Kewtox and hie con- 
temporaries very naturally desired to effect a similar solu- 
tion for the case of three bodies. The problem of motion 
in our solar system is that of the mutual action of a great 
number of bodies ; and having succeeded in the case of 
two bodies, it was necessary next to try that of three 



PROBLEMS OF GRAVITATION. 



141 



Thus arose the celebrated problem of three bodies. It is 
found that no rigorous and general solution of this problem 
is possible. The curves described by the several bodies 
would, in general, be so complex as to defy mathematical 
definition. But in the special case of motions in the solar 
system, the problem admits of being solved by approxima- 
tion with any required degree of accuracy. The princi- 
ples involved in this system of approximation may be com- 
pared to those involved in extracting the square root of 
any number which is not an exact square ; 2 for instance. 
The square root of 2 cannot be exactly expressed either 
by a decimal or vulgar fraction ; but by increasing the 
number of figures it can be expressed to any required limit 
of approximation. Thus, the vulgar fractions f , 1J, J-J|-, 
etc., are fractions which approach more and more to the 
required quantity ; and by using larger numbers the errors 
of such fraction may be made as small as we please. So, in 
using decimals, we diminish the error ten times for eve- 
ry decimal we add, but never reduce it to zero. A process 
of the same nature, but immensely more complicated, has 
to be used in computing the motions of the planets from 
their mutual gravitation. The possibility of such an ap- 
proximation arises from the fact that the planetary orbits 
are nearly circular, and that their masses are very small 
compared with that of the sun. The first approximation 
is that of motion in an ellipse. In this way the motion of 
a planet through several revolutions can nearly always be 
predicted within a small fraction of a degree, though it 
may wander widely in the course of centuries. Then sup- 
pose each planet to move in a known ellipse ; their mutual 
attraction at each point of their respective orbits can be 
expressed by algebraic formulae. In constructing these 
formulas, the orbits are first supposed to be circular ; and 
afterward account is taken by several successive steps of 
the eccentricity. Having thus found approximately their 
action on each other, the deviations from the pure elliptic 
motion produced by this action may be approximately cal- 



142 ASTRONOMY. 

culated. This being done, the motions will be more exact- 
ly determined, and the mutual action can be more exactly 
calculated. Thus, the process can be carried on step by 
step to any degree of precision ; but an enormous amount 
of calculation is necessary to satisfy the requirements of 
modern times with respect to precision.* As a general 
rule, every successive step in the approximation is much 
more laborious than all the preceding ones. 

To understand the principle of astronomical investiga- 
tion into the motion of the planets, the distinction be- 
tween observed and theoretical motions must be borne in 
mind. When the astronomer with his meridian circle de- 
termines the position of a planet on the celestial sphere, 
that position is an observed one. "Wlien he calculates it, for 
the same instant, from theory, or from tables founded on 
the theory, the result will be a calculated or theoretical 
position. The two are to be regarded as separate, no mat- 
ter if. they should be exactly the same in reality, because 
they have an entirely different origin. But it must be re- 
membered that no position can be calculated from theory 
alone independent of observation, because all sound theory 
requires some data to start with, which observation alone 
can furnish. In the case of planetary motions, these data 
are the elements of the planetary orbit already described, 
or, which amounts to the same tiling, the velocity and di- 
rection of the motion of the planet as well as its mass at 
some given time. If these quantities were once given 
with mathematical precision, it would be possible, from the 
theory of gravitation alone, without recourse to observa- 
tion, to predict the motions of the planets day by day 
and generation after generation with any required degree 
of precision, always supposing that they are subjected to no 
influence except their mutual gravitation according to the 
law of Newton. But it is impossible to determine the 
elements or the velocities without recourse to observation ; 

* In the works of the great mathematicians on this subject, algebraic 
formulae extending through many pages are sometimes given. 



PROBLEMS OF GRAVITATION. 143 

and however correctly they may seemingly be determined 
for the time being, subsequent observations always show 
them to have been more or less in error. The reader 
must understand that no astronomical observation can be 
mathematically exact. Both the instruments and the 
observer are subjected to influences which prevent more 
than an approximation being attained from any one 
observation. The great art of the astronomer consists in 
so treating and combining his observations as to eliminate 
their errors, and give a result as near the truth as possible. 
When, by thus combining his observations, the astrono- 
mer has obtained the elements of the planet's motion which 
he considers to be near the truth, he calculates from them 
a series of positions of the planet from day to day in the 
future, to be comj^ared with subsequent observations. If 
he desires his work to be more permanent in its nature, 
he may construct tables by which the position can be de- 
termined at any future time. Having thus a series of the- 
oretical or calculated places of the planet, he, or others, 
will compare his predictions with observation, and from 
the differences deduce corrections to his elements. We 
may say in a rough way that if a planet has been observed 
through a certain number of years, it is possible to calculate 
its place for an equal number of years in advance with 
some approach to precision. Accurate observations are 
commonly supposed to commence with Bradley, Astron- 
omer Royal of England in 1750. A century and a quarter 
having elapsed since that time, it is now possible to con- 
struct tables of the planets, which we may expect to be 
tolerably accurate, until the year 2000. But this is a 
possibility rather than a reality. The amount of calcu- 
lation required for such work is so immense as to be en- 
tirely beyond the power of any one person, and hence it is 
only when a mathematician is able to command the ser- 
vices of others, or when several mathematicians in some 
way combine for an object, that the best astronomical 
tables can hereafter be constructed. 



144 ASTRONOMY. 

§ 4. RESULTS OP GRAVITATION. 

From what we have said, it will be seen that the problem 
of the motions of the planets under the influence of grav- 
itation has called forth all the skill of the mathematicians 
who have attacked it. They actually find themselves able 
to reach a solution, which, so far as the mathematics of the 
subject are concerned, may be true for many centuries, but 
not a solution which shall be true for all time. Among 
those who have brought the solution so near to perfec- 
tion, La Place is entitled to the first rank, although there 
are others, especially La. G-kange, who are fully worthy to 
be named along with him. It will be of interest to state 
the general results reached by these and other mathema- 
ticians. 

We call to mind that but for the attraction of the 
planets upon each other, every planet would move around 
the sun in an invariable ellipse, according to Kepler's 
laws. The deviations from this elliptic motion produced 
"by their mutual attraction are called perturbations. When 
they were investigated, it was found that they were of two 
classes, which were denominated respectively periodic 
perturbations and secular variations. 

The periodic perturbations consist of oscillations depend- 
ent upon the mutual positions of the planets, and there- 
fore of comparatively short period. Whenever, after a 
number of revolutions, two planets return to the same 
position in their orbits, the periodic perturbations are of 
the same amount so far as these two planets are concerned. 
They may therefore be algebraically expressed as depend- 
ent upon the longitude of: the two planets, the disturbing 
one and the disturbed one. For instance, the perturba- 
tions of the earth produced by the action of Mercury 
depend on the longitude of the earth and on that of Mer- 
cury. Those produced by the attraction of Venus de- 
pend upon the longitude of the earth and on that of 
Venus, and so on. 



RESULTS OF GRAVITATION. 



145 



The secular perturbations, or secular variations as they 
are commonly called, consist of slow changes in the forms 
and positions of the several orbits. It is found that the 
perihelia of all the orbits are slowly changing their ap- 
parent directions from the sun ; that the eccentricities of 
some are increasing and of others diminishing ; and that 
the positions of the orbits are also changing. 

One of the first questions which arose in reference to 
these secular variations was, will they go on indefinitely ? 
If they should, they would evidently end in the subversion 
of the solar system and the destruction of all life upon the 
earth. The orbits of the earth and planets would, in the 
course of ages, become so eccentric, that, approaching 
near the sun at one time and receding far away from it at 
another, the variations of temperature would be destruc- 
tive to life. This problem was first solved by La Grange. 
He showed that the changes could not go on forever, but 
that each eccentricity would always be confined between 
two quite narrow limits. His results may be expressed 
by a very simple geometrical construction. Let S repre- 
sent the sun situated in the focus of the ellipse in which 




Pig. 54. 

the planet moves, and let G be the centre of the ellipse. 
Let a straight line SB emanate from the sun to B, 
another line pass from B to D, and so on ; the number of 
these lines being equal to that of the planets, and the last 
one terminating in O, the centre of the ellipse. Then the 
line S B will be moving around the sun with a very slow 
motion ; B D will move around B with a slow motion 
somewhat different, and so each one will revolve in the 



146 ASTRONOMY. 

same manner until we reach the line which carries on its 
end the centre of the ellipse. These motions are so slow 
that some of them require tens of thousands, and others 
hundreds of thousands of years to perform the revolution. 
By the combined motion of them all, the centre of the 
ellipse describes a somewhat irregular curve. It is evi- 
dent, however, that the distance of the centre from the 
sun can never be greater than the sum of these revolving 
lines. Now this distance shows the eccentricity of the 
ellipse, which is equal to half the difference between the 
greatest and least distances of the planet from the sun. 
The perihelion being in the direction C S, on the opposite 
side of the sun from O, it is evident that the motion of 
G will carry the perihelion with it. It is found in this 
way that the eccentricity of the earth's orbit has been 
diminishing for about eighteen thousand years, and will 
continue to diminish for twenty-five thousand years to 
come, when it will be more nearly circular than any orbit 
of our system now is. But before becoming quite circu- 
lar, the eccentricity will begin to increase again, and so go 
on oscillating indefinitely. 

Secular Acceleration of the Moon. — Another remark- 
able result reached by mathematical research is that of the 
acceleration of the moon's motion. More than a century 
ago it was found, by comparing the ancient and modern 
observations of the moon, that the latter moved around the 
earth at a slightly greater rate than she did in ancient 
times. The existence of this acceleration was a source of 
great perplexity to La Grange and La Place, because 
they thought that they had demonstrated mathematically 
that the attraction could not have accelerated or retarded 
the mean motion of the moon. But on continuing his in- 
vestigation, La Place found that there was one cause 
which he omitted to take account of — namely, the secular 
diminution in the eccentricity of the earth's orbit, of 
which we have just spoken. He found that this change 
in the eccentricity would slightly alter the action of the 



ACCELERATION OP THE MOON. 14? 

sun upon the moon, and that this alteration of action 
would be such that so long as the eccentricity grew 
smaller, the motion of the moon would continue to be ac- 
celerated. Computing the moon's acceleration, he found it 
to be equal to ten seconds into the square of the number 
of centuries, the law being the same as that for -the motion 
of a falling body. That is, while in one century she would 
be ten seconds ahead of the place she would have occupied 
had her mean motion been uniform, she would, in two 
centuries,' be forty seconds ahead, in three centuries ninety 
seconds, and so on ; and during the two thousand years 
which have elapsed since the observations of Hipparchus, 
the acceleration would be more than a degree. It has re- 
cently been found that La Place's calculation was not com- 
plete, and that with the more exact methods of recent times 
the real acceleration computed from the theory of gravita- 
tion is only about six seconds. The observations of ancient 
eclipses, however, compared with our modern tables, show 
an acceleration greater than this ; but owing to the rude 
and doubtful character of nearly all the ancient data, there 
is some doubt about the exact amount. From the most 
celebrated total eclipses of the sun, an acceleration of about 
twelve seconds is deduced, while the observations of 
Ptolemy and the Arabian astronomers indicate only eight 
or nine seconds. There is thus an apparent discrepancy 
between theory and observation, the latter giving a larger 
value to the acceleration. This difference is now accounted 
for by supposing that the motion of the earth on its axis 
is retarded — that is, that the day is gradually growing 
longer. From the modern theory of friction, it is found 
that the motion of the ocean under the influence of the 
moon's attraction which causes the tides, must be accom- 
panied with some friction, and that this friction must re- 
tard the earth's rotation. There is, however, no way of 
determining the amount of this retardation unless we 
assume that it causes the observed discrepancy between 
the theoretical and observed accelerations of the moon. 



148 ASTUONOMY. 

How this effect is produced will be seen by reflecting that 
if the day is continually growing longer without our know- 
ing it, our observations of the moon, which we may suppose 
to be made at noon, for example, will be constantly made a 
little later, because the interval from one noon to another 
will be continually growing a little longer. The moon con- 
tinually moving forward, the observation will place her fur- 
ther and further ahead than she would have been observed 
had there been no retardation of the time of noon. If in 
the course of ages our noon-dials get to be an hour too 
late, we should find the moon ahead of her calculated place 
by one hour's motion, or about a degree. The present 
theory of acceleration is, therefore, that the moon is really 
accelerated about six seconds in a century, and that the 
motion of the earth on its axis is gradually diminishing 
at such a rate as to produce an apparent additional ac- 
celeration which may range from two to six seconds. 



§ 5. REMARKS ON THE THEORY OF GRAVITA- 
TION. 

The real nature of the great discovery of Newton is so- 
frequently misunderstood that a little attention may be 
given to its elucidation. Gravitation is frequently spoken 
of as if it were a theory of Newton's, and very generally 
received by astronomers, but still liable to be ultimately 
rejected as a great many other theories have been. Not 
infrequently people of greater or less intelligence are 
found making great efforts to prove it erroneous. Every 
prominent scientific institution in the world frequently 
receives essays having this object in view. Now, the fact 
is that Newton did not discover any new force, but only 
showed that the motions of the heavens could be accounted 
for by a force which we all know to exist. Gravitation 
(Latin gravitas — weight, heaviness) is, properly speaking, 
the force which makes all bodies here at the surface of the 
earth tend to fall downward ; and if anv one wishes to 






REALITY OF GRAVITATION. 149 

-subvert the theory of gravitation, lie must begin by prov- 
ing that this force does not exist. This no one would 
think of doing. What Newton did was to show that 
this force, which, before his time, had been recognized 
only as acting on the surface of the earth, really extended 
to the heavens, and that it resided not only in the earth 
itself , but in the heavenly bodies also, and in each particle 
of matter, however situated. To put the matter in a terse 
form, what Newton discovered was not gravitation, but 
the universality of gravitation. 

It may be inquired, is the induction which supposes 
gravitation universal so complete as to be entirely beyond 
doubt ? We reply that within the solar system it certainly 
is. The laws of motion as established by observation and 
experiment at the surface of the earth must be considered 
as mathematically certain. Now, it is an observed fact 
that the planets in their motions deviate from straight 
lines in a certain way. By the first law of motion, such 
deviation can be produced only by a force ; and the direc- 
tion and intensity of this force admit of being calculated 
once that the motion is determined. When thus calculated, 
it is found to be exactly represented by one great force 
constantly directed toward the sun, and smaller subsidiary 
forces directed toward the several planets. Therefore, 
no fact in nature is more firmly established than is that of 
universal gravitation, as laid down by Newton, at least 
within the solar system. 

We shall find, in describing double stars, that gravita- 
tion is also found to act between the components of a great 
number of such stars. It is certain, therefore, that at 
least some stars gravitate toward each other, as the bodies 
of the solar system do ; but the distance which separates 
most of the stars from each other and from our sun is so 
immense that no evidence of gravitation between them 
has yet been given by observation. Still, that they do 
gravitate according to Newton's law can hardly be seri- 
ously doubted by any one who understands the subject. 



150 ASTRONOMY. 

The reader may now be supposed to see the absurdity of 
supposing that the theory of gravitation can ever be sub- 
verted. It is not, however, absurd to suppose that it may 
yet be shown to be the result of some more general law. 
Attempts to do this are made from time to time by men 
of a philosophic spirit ; but thus far no theory of the sub- 
ject having the slightest probability in its favor has been 
propounded. 

Perhaps one of the most celebrated of these theories is 
that of George Lewis Le Sage, a Swiss physicist of the 
last century. He supposed an infinite number of ultra- 
mundane corpuscles, of transcendent minuteness and veloc- 
ity, traversing space in straight lines in all directions. A 
single body placed in the midst of such an ocean of mov- 
ing corpuscles would remain at rest, since it would be equal- 
ly impelled in every direction. But two bodies would ad- 
vance toward each other, because each of them would 
screen the other from these corpuscles moving in the 
straight line joining their centres, and there would be a 
slight excess of corpuscles acting on that side of each 
body which was turned away from the other.* 

One of the commonest conceptions to account for grav- 
itation is that of a fluid, or ether, extending through all 
space, which is supposed to be animated by certain vibra- 
tions, and forms a vehicle, as it were, for the transmission 
of gravitation. This and all other theories of the kind 
are subject to the fatal objection of proposing complicated 
systems to account for the most simple and elementary 
facts. If, indeed, such systems were otherwise known to 
exist, and if it could be shown that they really would 
produce the effect of gravitation, they would be entitled 
to reception. But since they have been imagined only to 
account for gravitation itself, and since there is no proof 
of their existence except that of accounting for it, they 

* Reference may be made to an article on the kinetic theories of 
gravitation by William B. Taylor, in the Smithsonian Report for 
1876. 



CAUSE OF GRAVITATION. 151 

are not entitled to any weight whatever. In the present 
state of science, we are justified in regarding gravitation as 
an ultimate principle of matter, incapable of alteration by 
any transformation to which matter can be subjected. 
The most careful experiments show that no chemical pro- 
cess to which matter can be subjected either increases or 
diminishes its gravitating/principles in the slightest degree. 
We cannot therefore see how this principle can ever be 
referred to any more general cause. 



CHAPTER VI. 

THE MOTIONS AND ATTRACTION OF THE MOON. 

Each of the planets, except Mercury and Venus, is at- 
tended by one or more satellites, or moons as they are some- 
times familiarly called. These objects revolve around their 
several planets in nearly circular orbits, accompanying them 
in their revolutions around the sun. Their distances from 
their planets are very small compared with the distances 
of the latter from each other and from the sun. Their 
magnitudes also are very small compared with those of the 
planets around which they revolve. Where there are 
several satellites revolving around a planet, the whole of 
these bodies forms a small system similar to the solar sys- 
tem in arrangement. Considering each system by itself, 
the satellites revolve around their central planets or 
" primaries," in nearly circular orbits, much as the planets 
revolve around the sun. But each system is carried around 
the sun without any serious derangement of the motion 
of its several bodies among themselves. 

Our earth has a single satellite accompanying it in this 
way, the familiar moon. It revolves around the earth in 
a little less than a month. The nature, causes and con- 
sequences of this motion form the subject of the present 
chapter. 

§ 1. THE MOON'S MOTIONS AND PHASES. 

That the moon performs a monthly circuit in the heav- 
ens is a fact with which we are all familiar from child- 
hood. At certain times we see her newly emerged from 



MOTION OF THE MOON. 153 

the sun's rays in the western twilight, and then we call 
her the new moon. On each succeeding evening, we see 
her further to the east, so that in two weeks she is oppo- 
site the sun, rising in the east as he sets in the west. 
Continuing her course two weeks more, she has approached 
the sun on the other side, or from the west, and is once 
more lost in his rays. At the end of twenty-nine or thirty 
days, we see her again emerging as new moon, and her cir- 
cuit is complete. It is, however, to be remembered 
that the sun lias been apparently moving toward the east 
among the stars during the whole month, so that during 
the interval from one new moon to the next the moon has 
to make a complete circuit relatively to the stars, and 
move forward some 30° further to overtake the sun. The 
revolution of the moon among the stars is performed in 
about 2TJ days,* so that if we observe when the moon is 
very near some star, we shall find her in the same position 
relative to the star at the end of this interval. 

The motion of the moon in this circuit differs from the 
apparent motions of the planets in being always forward. 
We have seen that the planets, though, on the whole, mov- 
ing directly, or toward the east, are affected with an ap- 
parent retrograde motion at certain intervals, owing to the 
motion of the earth around the sun. But the earth is the 
real centre of the moon's motion, and carries the moon 
along with it in its annual revolution around the sun. To 
form a correct idea of the real motion of these three 
bodies, we must imagine the earth performing its circuit 
around the sun in one year, and carrying with it the moon, 
which makes a revolution around it in 27 days, at a distance 
only about T fo that of the sun. 

In Fig. 55 suppose S to represent the sun, the large 
circle to represent the orbit of the earth around it, E to 
be some position of the earth, and the dotted circle to rep- 
resent the orbit of the moon around the earth. We must 

* More exactly, 27-33166 d . 



I 



154 



ASTRONOMY. 




Fro. 55. 



imagine the latter to cany this circle with it in its an- 
nual course around the sun. Suppose that when the earth 
is at E the moon is at M. Then if the earth move to 

J£ x in 27 J days, the moon 
will have made a complete 
revolution relative to the 
stars — that is, it will be at 
M ' 2 , the line E l J/" a being par- 
allel to E M. But new 
moon will not have arrived 
again because the sun is not 
in the same direction as be- 
fore. The moon must move 
through the additional arc 
M l EJf. 2 , and a little more, 
owing to the continual ad- 
vance of the earth, before it 
will again be new moon. 
Phases of the Moon. — The moon being a non-luminous 
body shines only by reflecting the light falling on her 
from some other body. The principal source of light is 
the sun. Since the moon is spherical in shape, the sun 
can illuminate one half her surface. The appearance of 
the moon varies according to the amount of her illumi- 
nated hemisphere which is turned toward the earth, as 
can be seen by studying Fig. 56. Here the central 
globe is the earth ; the circle around it represents the orbit 
of the moon. The rays of the sun fall on both earth and 
moon from the right, the distance of the sun being, on the 
scale of the figure, some 30 feet. Eight positions of the 
moon are shown around the orbit at A, E, C, etc., and 
the right-hand hemisphere of the moon is illuminated in 
each position. Outside these eight positions are eight 
others showing how the moon looks as seen from the earth 
in each position. 

At A it is " new moon," the moon being nearly 
between the earth and the sun. Its dark hemisphere 



PHASES OF THE MOON. 



155 



is then turned toward the earth, so that it is entirely 
invisible. 

At E the observer on the earth sees about a fourth of 
the illuminated hemisphere, which looks like a crescent, 
as shown in the outside figure. In this position a great 
deal of light is reflected from the earth to the moon, ren- 
dering the dark part of the latter visible by a gr;iy light. 




Fig. 56. 



This appearance is sometimes called the " old moon in 
the new moon's arms." 

At 6 7 the moon is said to be in her " first quarter," and 
one half her illuminated hemisphere is visible. 

At G three fourths of the illuminated hemisphere is 
visible, and at B the whole of it. The latter position, when 
the moon is opposite the sun, is called " full moon." 

After this, at H, D, F, the same appearances are re- 
peated in the reversed order, the position I) being called 
the "last quarter." 



156 ASTRONOMY. 

The four principal phases of the moon are, " New 
moon," " First quarter," " Full moon," " Last quarter," 
which occur in regular and unending succession, at inter- 
vals of between 7 and 8 days. 

§2. THE SUN'S DISTURBING FORCE. 

The distances of the sun and planets being so immensely 
great compared with that of the moon, their attraction 
upon the earth and the moon is at all times very nearly 
equal. Now it is an elementary principle of mechanics 
that if two bodies are acted upon by equal and parallel 
forces, no matter how great these forces may be, the 
bodies will move relatively to each other as if those forces 
did not act at all, though of course the absolute motion of 
each will be different from ..what it otherwise would be. 
If we calculate the absolute attraction of the sun upon the 
moon, we shall find it to be about twice as great as that of 
the earth, because, although it is situated at 400 times the 
distance, its mass is about 330,000 times as great as that of 
the earth, and if we divide this mass by the square of the 
distance 400 we have 2 as the quotient. 

To those unacquainted with mechanics, the difficulty 
often suggests itself that the sun ought to draw the moon 
away from the earth entirely. But we are to remember 
that the sun attracts the earth in the same way that it at- 
tracts the moon, so that the difference between the sun's 
attraction on the moon and on the earth is only a small 
fraction of the attraction between the earth and the moon.' 5 * 
As a consequence of these forces, the moon moves around 
the earth nearly as if neither of them were attracted by 

* In this comparison of the attractive forces of the sun upon the 
moon and upon the earth, the reader will remember that we are speak- 
ing not of the absolute force, but of what is called the accelerating force, 
which is properly the ratio of the absolute force to the mass of the 
body attracted. Th : earth having 80 times the mass of the moon, the 
sun must of course attract it with 80 times the absolute force in order 
to produce the same motion, or the same accelerating force. 



SUN'S ATTRACTION ON MOON. 157 

the sun — that is, nearly in an ellipse, having the earth in 
its focus. But there is always a small difference between 
the attractive forces of the sun upon the moon and upon the 
earth, and this difference constitutes a disturbing force 
which makes the moon deviate from the elliptic orbit 
which it would otherwise describe, and, in fact, keeps the 
ellipse which it approximately describes in a state of con- 
stant change. 

A more precise idea of the manner in which the sun disturbs the 
motion of the moon around the earth may be gathered from 
Fig. 57. Here 8 represents the sun, and the circle F Q M ^repre- 
sents the orbit of the moon. First suppose the moon at N, the posi- 
tion corresponding to new moon. Then the moon, being nearer to 
the sun than the earth is, will be attracted more powerfully by it 
than the earth is. It will therefore be drawn away from the earth, 
or the action of the sun will tend to separate the two bodies. 




Fig. 57. 

Next suppose the moon at F the position corresponding to full 
moon. Here the action of the sun upon the earth will be more 
powerful than upon the moon, and the earth will in consequence be 
drawn away from the moon. In this position also the effect of the 
disturbing force is to separate the two bodies. If, on the other 
hand, the moon is near the first quarter or near Q, the sun will exert 
a nearly equal attraction on both bodies ; and ince the lines of at- 
traction E S and Q S then converge toward S, it follows that there 
will be a tendency to bring the two bodies together. The same 
will evidently be true at the third quarter. Hence the influence of 
the disturbing force changes back and forth twice in the course of 
each lunar month. 

The disturbing force in question may be constructed for any po- 
sition of the moon in its orbit in the following way, which is be- 
lieved to be due to Mr. R. A. Proctor : Let J/" be the position of 
the moon ; let us represent the sun's attraction upon it by the line 
M S, and let us investigate what line will represent the sun's attrac- 
tion upon the earth on the same scale. From if drop the perpen- 



158 ASTRONOMY. 

dicular MP upon the line E 8 joining the sun to the earth. This 
attraction being inversely as the square of the distance, we shall 
have, 

Attraction on earth _ 8 M" 1 

Attraction on moon ~~ S E r 

We have taken the line S M itself to represent the attraction on 
the moon, so that we have 

Attraction on moon = S M. 

Multiplying the two equations member by member, we find, 

S M 2 
Attraction on earth = 8 M x ■ 

o hi 

The line S M is nearly equal to 8 P, so that we may take for an 
approximation to the required line, 

sp x ?4^ = sp x _ — _ = sp 



SE' 1 {SP+PEf / PE 



/ PE\- 

i} + sp) 



= £P(l_2^|+etc.), 



the last equation being obtained by the binomial theorem. But 

PE 
the fraction ^-=, is so small, being less than T ^, that its powers 

above the first will be small enough to be neglected. So we shall 
have for the required line, 

SP-2EP. 

If, therefore, we take the point A so that P A shall be equal to 2 
E P, the attraction of the sun upon the earth will on the same scale be 
represented by the line A S. The disturbing force which we seek 
is represented by the difference between the attraction of the sun 
upon the earth and that of the same body upon the moon. If then 
we suppose the force A S to be applied to the moon in the opposite 
direction, the resultant of the two forces M S and S A will repre- 
sent the disturbing force required. By the law of the composition 
of forces, this resultant is represented by the line MA. 

We are thus enabled to construct this force in a very simple man- 
ner, when the moon is in any given position. When the moon is 
at N, the line JSf A will be equal to 2 EM; the disturbing force 
will therefore be represented by twice the distance of the moon. 
On the other hand, vs r hen the moon is at Q the three points E jV 
and A will all coincide. Hence the disturbing force which tends 
to bring the moon toward the earth will be represented by the line 
Q E ; hence the force which tends to draw r the moon away from the 
earth at new and full moon is twice as great as that which draws 



MOON'S NODES. 159 

the bodies together at the quarters. Consequently, upon the whole, 
the tendency of the sun's attraction is to diminish the attraction of 
the earth upon the moon. 

§ 3. MOTION OF THE MOON'S NODES. 

Among the changes which the sun's attraction produces 
in the moon's orbit, that which interests us most is the 
constant variation in the plane of the orbit. This plane 
is indicated by the path which the moon seems to describe 
in its circuit around the celestial sphere. Simple naked 
eye estimates of the moon's position, continued during a 
month, would show that her path was always quite near 
the ecliptic, because it would be evident to the eye that, 
like the sun, she was much farther north while passing 
from the vernal to the autumnal equinox than while de- 
scribing the other half of her circuit from the autumnal 
to the vernal equinox. It would be seen that, like the 
sun, she was farthest north in about six hours of right as- 
cension, and farthest south when in about eighteen hours 
of right ascension. 

To map out the path with greater precision, we have to 
observe the position of the moon from night to night with 
a meridian circle. We thus lay down her course among 
the stars in the same manner that we have formerly shown 
it possible to lay down the sun's path, or the ecliptic. It 
is thus found that the path of the moon may be considered 
as a great circle, making an angle of 5° with the ecliptic, 
and crossing the ecliptic at this small angle at two oppo- 
site points of the heavens. These points are called the 
moon's nodes. The point at which she passes from the 
south to the north of the ecliptic is called the ascending 
node / that in which she passes from the north to the 
south is the descending node. To illustrate the motion of 
the moon near the node, the dotted line a a may be taken 
as showing the path of the moon, while the circles show 
her position at successive intervals of one hour as she is ap- 
proaching her ascending node. Position number 9 is exactly 



160 



ASTRONOMY. 




at the node. If we 
continue following her 
course in this way for 
a week, we should find 
that she had moved 
about 90°, and attained 
her greatest north lati- 
tude at 5° from the 
ecliptic. At the end 
of another week, we 
should find that she- 
had returned to the 
ecliptic and crossed it 
at her descending node. 
At the end of the third 
week very nearly, we 
should find that she had 
made three fourths the 
circuit of the heavens, 
and was now in her 
greatest south latitude, 
being 5° south of the 
ecliptic. At the end 
of six or seven days 
more, we should again 
find her crossing the 
ecliptic at her ascend- 
ing node as before. We 
may thus conceive of 
four cardinal points of 
the moon's orbit, 90°" 
apart, marked by the 
two nodes and the two- 
points of greatest north 
and south latitude. 

Motion of the Nodes. 
— A remarkable prop- 



MOON'S NODES. 101 

erty of these points is that they are not fixed, but are con- 
stantly moving. The general motion is a little irregular, 
but, leaving out small irregularities, it is constantly toward 
the west. Thus returning to our watch of the course of 
the moon, we should lind that, at her next return to the 
ascending node, she would not describe the line a a as 
before, but the line h h about one fourth of a diameter 
north of it. She would therefore reach the ecliptic more 
than 1^° west of the preceding point of crossing, and her 
other cardinal points would be found 1-J° farther west as 
she went around. On her next return she would describe 
the line c c, then the line d d, etc., indefinitely, each line 
being farther toward the west. The figure shows the 
paths in five consecutive returns to the node. 

A lapse of nine years will bring the descending node 
around to the place which was before occujued by the 
ascending node, and thus we shall have the moon crossing 
at a small inclination toward the south, as shown in the 
figure. 

A complete revolution of the nodes takes place in IS -6 
years. After the lapse of this period, the motion is re- 
peated in the same manner. 

One consequence of this motion is that the moon, after 
leaving a node, reaches the same node again sooner than 
she completes her true circuit in the heavens. How much 
sooner is readily computed from the fact that the retro- 
grade motion of the node amounts to 1° 26' 31" during 
the period that the moon is returning to it. It takes the 
moon about two hours and a half (more exactly d .10944) 
to move through this distance ; consequently, comparing 
w r ith the sidereal period already given, we find that the 
return of the moon to her node takes place in 27 d .32166 
— () d . 10944 = 27 d .21222. This time will be important to 
us in considering the recurrence of eclipses. 

In Fig. 59 is illustrated the effect of these changes in 
the position of the moon's orbit upon her motion rela* 






162 



ASTRONOMY. 



tive to the equator. E here represents the vernal and 

A the autumnal equinox, situated 
180° apart. In March, 1876, 
the moon's ascending node cor- 
responded with the vernal equi- 
nox, and her descending node 
with the autumnal one. Conse- 
quently she was 5° north of the 
ecliptic when in six hours of 
right ascension or near the mid- 
dle of the figure. Since the 
ecliptic is 23^° north of the 
equator at this point, the moon at- 
tained a maximum declination of 
28£° ; she therefore passed nearer 
the zenith when in six hours 
of right ascension than at any 
other time during the eighteen 
years' period. In the language 
of the almanac, " the moon ran 
high." Of course when at her 
greatest distance south of the 
equator, in the other half of her 
orbit, she attained a correspond- 
ing south declination, and cul- 
minated at a lower altitude than 
she had for eighteen years. In 
1885 the nodes will change places, 
and the orbit will deviate from 
the equator less than at any other 
time during the eighteen years. 
In 1880 the descending node will 
be in six hours of right ascension, 
and the greatest angular distance 
of the moon from the equator 

will be nearly equal to that of the sun. 




PERIGEE OF THE MOON. 163 

§ 4. MOTION OF THE PERIGEE. 

If the sun exerted no disturbing force on the moon, the 
latter would move round the earth in an ellipse according 
to Kepler's laws. But the difference of the sun's attrac- 
tion on the earth and on the moon, though only a small 
fraction of the earth's attractive force on the moon, is yet 
so great as to produce deviations from the elliptic motion 
very much greater than occur in the motions of the planets. 
It also produces rapid changes in the elliptic orbit. The 
most remarkable of these changes are the progressive 
motion of the nodes just described and a corresponding 
motion of the perigee. Referring to Fig. 52, which illus- 
trated the elliptic orbit of a planet, let us suppose it to 
represent the orbit of the moon. S will then represent 
the earth instead of the sun, and n will be the lunar per- 
igee, or the point of the orbit nearest the earth. But, 
instead of remaining nearly fixed, as do the orbits of the 
planets, the lunar orbit itself may be considered as making 
a revolution round the earth in about nine years, in the 
same direction as the moon itself. Hence if we note the 
longitude of the moon's perigee at any time, and again 
two or three years later, we shall find the two positions 
quite different. If we wait four years and a half, we shall 
find the perigee in directly the opposite point of the 
heavens. 

The eccentricity of the moon's orbit is about 0.055, and 
in consequence the moon is about 6° ahead of its mean 
place when 90° past the perigee, and about the same dis- 
tance behind when half way from apogee to perigee. 

The disturbing action of the sun produces a great num- 
ber of other inequalities, of which the largest are the 
evection and the variation. The former is more than a 
degree, and the latter not much less. The formulae by 
which they are expressed belong to Celestial Mechanics, 
and the reader who desires to study them is referred to 
works on that subject. 






104 ASTRONOMY. 



§ 5. ROTATION OF THE MOON. 

The moon rotates on her axis in the same time and in 
the same direction in which she revolves around the earth. 
In consequence she always presents very nearly the same 
face to the earth.* There is indeed a small oscillation 
called the libration of the moon, arising from the fact that 
her rotation on her axis is uniform, while her revolution 
around the earth is not uniform. In consequence of 
this we sometimes see a little of her farther hemisphere 
first on one side and then on the other, but the greater 
part of this hemisphere is forever hidden from human 
sight. 

The axis of rotation of the moon is inclined to the 
ecliptic about 1° 29'. It is remarkable that this axis 
changes its direction in a way corresponding exactly to 
the motion of the nodes of the moon's orbit. Let us sup- 
pose a line passing through the centre of the earth per- 
pendicular to the plane of the moon's orbit. In conse- 
quence of the inclination of the orbit to the ecliptic, this 
line will point 5° from the pole of the ecliptic. Then, 
suppose another line parallel to the moon's axis of rota- 
tion. This line will intersect the celestial sphere 1° 29' 
from the pole of the ecliptic, and on the opposite side 
from the pole of the moon's orbit, so that it will be 6^-° 
from the latter. As one pole revolves around the 
pole of the ecliptic in 18.6 years, the other will do the 
same, always keeping the same position relative to the 
first. 



* This conclusion is often a pons asinorum to some who conceive 
that, if the same face of the moon is always presented to the earth, she 
cannot rotate at all. The difficulty arises from a misunderstanding of 
the difference between a relative and an absolute rotation. It is true 
that she does not rotate relatively to the line drawn from the earth to 
her centre, but she must rotate relative to a fixed line, or a line drawn 
to a fixed star. 



THE TIDES. 105 



§ 6. THE TIDES. 

The ebb and flow of the tides are produced by the un- 
equal attraction of the sun and moon on different parts of 
the earth, arising from the fact that, owing to the magni- 
tude of the earth, some ]3arts of it are nearer these attracting 
bodies than others, and are therefore more strongly at- 
tracted. To understand the nature of the tide-producing 
force, we must recall the principle of mechanics already 
cited, that if two neighboring bodies are acted on by 
equal and parallel accelerating forces, their motion rel- 
ative to each other will not be altered, because both will 
move equally under the influence of the forces. When 
the forces are slightly different, either in magnitude or 
direction or both, the relative motion of the two bodies 
will depend on this difference alone. Since the sun and 
moon attract those parts of the earth which are nearest 
them more powerfully than those which are remote, there 
arises an inequality which produces a motion in the 
waters of the ocean. As the earth revolves on its axis, 
different parts of it are brought in succession under the 
moon. Thus a motion is produced in the ocean, which 
goes through its rise and fall according to the apparent 
position of the moon. This is called the tidal wave. 

The tide-producing force of the sun and moon is so nearly like 
the disturbing force of the sun upon the motion of the moon around 
the earth that nearly the same explanation will apply to both. Let 
us then refer again to Fig. 57, and suppose E to represent the 
centre of the earth, the circle F Q N its circumference, M a par- 
ticle of water on the earth's surface, and 8 either the sun or the 
moon. 

The entire earth being rigid, each part of it will move under the 
influence of the moon's attraction as if the whole were concen- 
trated at its centre. But the attraction of the moon upon the 
particle M, being different from its mean attraction on the earth, will 
tend to make it move differently from the earth. The force which 
causes this difference of motion, as already explained, will be repre- 
sented by the line M A. It is true that this same disturbing force is 
acting upon that portion of the solid earth at M as well as upon the 
water. But the earth cannot yield on account of its rigidity ; the 



166 ASTRONOMY. 

water therefore tends to flow along the earth's surface from M 
toward N. There is therefore a residual force tending to make the 
water higher at N than at M. 

If we suppose the particle M to be near F, then the point A will 
be to the left of F. The water will therefore be drawn in an oppo- 
site direction or toward F. There will therefore also be a force 
tending to make the water accumulate around F. As the disturb- 
ing force of the sun tends to cause the earth and moon to separate 
both at new and full moon, so the tidal force of the sun and 
moon upon the earth tends to make the waters accumulate both at 
M and F. More exactly, the force in question tends to draw the 
earth out into the form of a prolate ellipsoid, having its longest 
axis in the direction of the attracting body. As the earth rotates 
on its axis, each particle of the ocean is, in the course of a day, 
brought in to the four positions N Q F R, or into some positions 
corresponding to these. Tims, the tide-producing force changes 
back and forth twice in the course of a lunar day. (By a lunar day 
we mean the interval between two successive passages of the moon 
across the meridian, which is, on the average, about 24 h 48 m .) If the 
waters could yield immediately to this force, we should always have 
high tide at F and N and low tides at Q and R. But there are two 
causes which prevent this. 

1. Owing to the inertia of the water, the force must act some 
time before the full amount of motion is produced, and this motion, 
once attained, will continue after the force has ceased to act. 
Again, the waters will continue to accumulate as lcng as there is 
any motion in the required direction. The result of this would be 
high tides at Q and R and low tides at F and JV, if the ocean 
covered the earth and were perfectly free to move. That is, high 
tides would then be six hours after the moon crossed the meridian. 

2. The principal cause, however, which interferes with the 
regularity of the motion is the obstruction of islands and continents 
to the free motion of the water. These deflect the tidal wave from 
its course in so many different ways, that it is hardly possible to 
trace the relation between the attraction of the moon and the mo- 
tion of the tide ; the time of high and low tide must therefore be 
found by observing at each point along the coast. By comparing 
these times through a series of years, a very accurate idea of the 
motion of the tidal wave can be obtained. 

Such observations have been made over our Atlantic aud Pacific 
coasts by the Coast Survey and over most of the coasts of Europe, 
by the countries occupying them. Unfortunately the tides cannot 
be observed away from the land, and hence little is known of the 
course of the tidal wave over the ocean. 

We have remarked that both the sun and moon exert a 
tide-producing force. That of the sun is about T 4 ¥ of that 
of the moon. At new and full moon the two forces are 
united, and the actual force is equal to their sum. At 



THE TIDES. 167 

first and last quarter, when the two bodies are 90° apart, 
they act in opposite directions, the sun tending to produce 
a high tide where the moon tends to produce a low one, 
and vice versa. The result of this is that near the time of 
new and full moon we have what are known as the spring 
tides, and near the quarters what are called neap tides. If 
the tides were always proportional to the force which pro- 
duces them, the sjuing tides would be highest at full 
moon, but the tidal wave tends to go on for some time 
after the force which produces it ceases. Hence the high- 
est spring tides are not reached until two or three days after 
new and full moon. Again, owing to the effect of fric- 
tion, the neap tides continue to be less and less for two or 
three days after the first and last quarters, when the grad- 
ually increasing force again has time to make itself felt. 

The theory of the tides offers very complicated prob- 
lems, which have taxed the powers of mathematicians for 
several generations. These problems are in their elements 
less simple than those presented by the motions of the 
planets, owing to the number of disturbing circumstances 
which enter into them. The various depths of the ocean 
at different points, the friction of the water, its momen- 
tum when it is once in motion, the effect of the coast-lines, 
have all to be taken into account. These quantities are 
so far from being exactly known that the theory of the 
tides can be expressed only by some general principles 
which do not suffice to enable us to predict them for any 
given place. From observation, however, it is easy to 
construct tables showing exactly what tides correspond to 
given positions of the sun and moon at any port where the 
observations are made. With such tables the ebb and now 
are predicted for the benefit of all who are interested, but 
the results may be a little uncertain on account of the 
effect of the winds upon the motion of the water. 



CHAPTER VII. 

ECLIPSES OF THE SUN AND MOON. 

Eclipses are a class of phenomena arising from the 
shadow of one body being cast upon another, or from a dark 
body passing over a bright one. In an eclipse of the sun, 
the shadow of the moon sweeps over the earth, and the 
sun is wholly or partially obscured to observers on that 
part of the earth where the shadow falls. In an eclipse of 
the moon, the latter enters the shadow of the earth, and is 
wholly or partially obscured in consequence of being de- 
prived of some or all its borrowed light. The satellites 
of other planets are from time to time eclipsed in the 
same way by entering the shadows of their primaries ; 
among these the satellites of Jupiter are objects whose 
eclipses may be observed with great regularity. 

§1. THE EARTH'S SHADOW AND PENUMBRA. 

In Fig. 60 let S represent the sun and E the earth. 
Draw straight lines, D B V and D' B' V, each tangent 
to the sun and the earth. The two bodies being supposed 
spherical, these lines will be the intersections of a cone 
with the plane of the paper, and may be taken to repre- 
sent that cone. It is evident that the cone B V B' will 
be the outline of the shadow of the earth, and that within 
this cone no direct sunlight can penetrate. It is therefore 
called the earth's shadow cone. 

Let us also draw the lines D f B P and D B' P' to rep- 
resent the other cone tangent to the sun and earth. It is- 



THE EARTH'S SHADOW. 



169 



then evident that within the region V B P and V B' P' 
the light of the sun will be partially but not entirely cut 
off. 




Fig. 60. — form of shadow. 

Dimensions of Shadow. —Let us investigate the distance E Ffrom 
the centre of the earth to the vertex of the shadow. The triangles 
V E B and V S D are similar, having a right angle at B and at D. 
Hence, 

YE: EB = VS: SD = ES: (SD — E B). 
So if we put 

l=VE, the length of the shadow measured from the centre of 
the earth. 

r = ES, the radius vector of the earth, 

12 = SD, the radius of the sun, 

p = E B, the radius of the earth, 

S, the angular semi-diameter of the sun as seen from the earth, 

it, the horizontal parallax of the sun, 

we have 



1= VE 



ES x EB 



rp 



SD-EB JR-f 
But by the theory of parallaxes (Chapter L, § V), 



Hence, 



p = r sin -k 
B = r sin S 

2 = 



sin aS — sin it 



The mean value of the sun's angular semi-diameter, from which 
the real value never differs by more than the sixtieth part, is found 
by observations to be about 16' 0" = 960", while the mean value of n 



170 ASTRONOMY. 

is about 8" • 8. We find sin S— sin tt = • 00461, and . n 9 . = 

sin 6 — sin x 

•ooitt = 217. We therefore conclude that the mean length of 
the earth's shadow is 217 times the earth's radius ; in round 
numbers 1,380,000 kilometres, or 800,000 miles, the mean radius 
of the earth being 6370 kilometres. It will be seen from the figure 
that it varies directly as the distance of the earth from the 
sun ; it is therefore about one sixtieth less than the mean in Decem- 
ber, and one sixtieth greater in June. 

The radius of the shadow diminishes uniformly with the distance 
as we go outward from the earth. At any distance z from the 



rmuia gives 



earth's centre it will be equal to [ 1 — - )p, for this fo 

the radius p when 2 = 0, and the diameter zero when z = I as it 
should.* 



§ 2. ECLIPSES OF THE MOON. 

The mean distance of the moon from the earth is about 
60 radii of the latter, while, as we have just seen, the 
length E V of the earth's shadow is 217 radii of the earth. 
Hence when the moon passes through the shadow she does 
so at a point less than three tenths of the way from 
E to V. The radius of the shadow here will be ^^P- 
of the radius E B of the earth, a quantity which we read- 
ily find to be about 4600 kilometres. The radius of the 
moon being 1736 kilometres, it will be entirely enveloped 
by the shadow when it passes through it within 2864 
kilometres of the axis E V of the shadow. If its least dis- 
tance from the axis exceed this amount, a portion of the 
lunar globe will be outside the limits B V of the shadow 
cone, and will therefore receive a portion of the direct 
light of the sun. If the least distance of the centre of the 
moon from the axis of the shadow is greater than the 
sum of the radii of the moon and the shadow — that is, 
greater than 6336 kilometres — the moon will not enter the 

* It will be noted that this expression is not, rigorously speaking, the 
semi-diameter of the shadow, but the shortest distance from a point on 
its central line to its conical surface. This distance is measured in a 
direction E B perpendicular to D B, whereas the diameter would be 
perpendicular to the axis 8 E, and its half length would be a little 
greater than E B. 



ECLIP8E8 OF THE MOON. 171 

shadow at all, and there will be no eclipse proper, though 
the brilliancy of the moon must be diminished wherever 
she is within the penumbral region. 

When an eclipse of the moon occurs, the phases are laid 
down in the almanac in the following manner : Supposing 
the moon to be moving around the earth from below up- 
ward, its advancing edge first meets the boundary B' P' 
of the penumbra. The time of this occurrence is given in 
the almanac as that of " moon entering penumbra." A 
small portion of the sunlight is then cut off from the ad- 
vancing edge of the moon, and this amount constantly in- 
creases until the edge reaches the boundary B' V of the 
shadow. It is curious, however, that the eye can scarcely 
detect any diminution in the brilliancy of the moon until 
she has almost touched the boundary of the shadow. The 
observer must not therefore expect to detect the coming 
eclipse until very nearly the time given in the almanac as 
that of ' ' moon entering shadow. ' ' As this happens, the 
advancing portion of the lunar disk will be entirely lost to 
view, as if it were cut off by a rather ill-defined line. It 
takes the moon about an hour to move over a distance 
equal to her own diameter, so that if the eclipse is nearly 
central the whole moon will be immersed in the shadow 
about an hour after she first strikes it. This is the time of 
beginning of total eclipse. So long as only a moderate 
portion of the moon's disk is in the shadow, that portion 
will be entirely invisible, but if the eclipse becomes total 
the whole disk of the moon will nearly always be plainly 
visible, shining with a red coppery light. This is owing to 
the refraction of the sun's rays by the lower strata of the 
earth's atmosphere. We shall see hereafter that if a ray of 
light D B passes from the sun to the earth, so as just to 
graze the latter, it is bent by refraction more than a de- 
gree out of its course, so that at the distance of the moon 
the whole shadow is filled with this refracted light. An 
observer on the moon would, during a total eclipse of the 
latter, see the earth surrounded by a ring of light, and this 



172 ASTRONOMY. 

ring would appear red, owing to the absorption of the blue 
and green rajs by the earth's atmosphere, just as the sun 
seems red when setting. 

The moon may remain enveloped in the shadow of the 
earth during a period ranging from a few minutes to nearly 
two hours, according to the distance at which she passes 
from the axis of the shadow and the velocity of her angu- 
lar motion. When she leaves the shadow, the phases 
which we have described occur in reverse order. 

It very often happens that the moon passes through the 
penumbra of the earth without touching the shadow at all. 
No notice is taken of these passages in our almanacs, be- 
cause, as already stated, the diminution of light is scarcely 
perceptible unless the moon at least grazes the edge of the 
shadow. 

§ 3. ECLIPSES OP THE SUN. 

In Fig. 60 we may suppose B E B' to represent the 
moon as well as the earth. The geometrical theory of the 
shadow will remain the same, though the length of the 
shadow will be much less. We may regard the mean semi- 
diameter of the sun as seen from the moou, and its mean 
distance, as being the same for the moon as for the earth. 
Therefore, in the formula which gives the length of the 
moon's shadow, S may retain the same value, while p and 
n must be diminished in the ratio of the moon's radius to 
that of the earth. The denominator, sin S— sin n, will be 
but slightly altered. The radius of the moon is about 1736 
kilometres. Multiplying this by 217, as before, we find 
the mean length of the moon's shadow to be 377,000 
kilometres. This is nearly equal to the distance of the 
moon from the earth when she is in conjunction with the 
sun. We therefore conclude that when the moon passes 
between the earth and the sun, the former will be very 
near the vertex V oi the shadow. As a matter of fact, 
an observer on the earth's surface will sometimes pass 



THE MOON'S SHADOW. 173 

through the region C VC, and sometimes on the other 
side of Y. 

Now, in Fig. 60, still supposing B E B' to be the 
moon, let us draw the lines D B' P' and D' B P tan- 
gent to both the moon and the sun, but crossing each other 
between these bodies at b. It is evident that outside the 
space P B B' P' an observer will see the whole sun, no 
part of the moon being projected upon it ; while within 
this space the sun will be more or less obscured. The 
whole obscured space may be divided into three regions, in 
each of which the character of the phenomenon is differ- 
ent from what it is in the others. 

Firstly, we have the region B V B' forming the shadow 
cone proper. Here the sunlight is entirely cut off by the 
moon, and darkness is therefore complete, except so far as 
light may enter by refraction or reflection. To an observer 
at V the moon would exactly cover the sun, the two 
bodies being apparently tangent to each other all around. 

Secondly, we have the conical region to the right of V 
between the lines B T^and B' V continued. In this 
region the moon is seen wholly projected upon the sun, 
the visible portion of the latter presenting the form of a 
ring of light around the moon. This ring of light will be 
wider in proportion to the apparent diameter of the sun, 
the farther out we go, because the moon will appear 
smaller than the sun, and its angular diameter will dimin- 
ish in a more rapid ratio than that of the sun. This 
region is that of annular eclipse, because the sun will pre- 
sent the appearance of an annulus or ring of light around 
the moon. 

Thirdly, we have the region P B V and P' B' V, which 
we notice is connected, extending around the interior cone. 
An observer here would see the moon partly projected 
upon the sun, and therefore a certain part of the sun's 
light would be cut off. Along the inner boundary B V 
and B r V the obscuration of the sun will be complete, 
but the amount of sunlight will gradually increase out to 



174 



ASTRONOMY. 



the outer boundary B P B' P\ where the whole sun in 
visible. This region of partial obscuration is called the 
penumbra. 

To show more clearly the phenomena of solar eclipse, 
we present another figure representing the penumbra of 




Fig. 61. — figure of shadow for anisttlar eclipse. 

the moon thrown upon the earth.* The outer of the two 
circles S represents the limb of the sun. The exterior tan- 
gents which mark the boundary of the shadow cross each 
other at V before reaching the earth. The earth being 
a little beyond the vertex of the shadow, there can be no 
total eclipse. In this case an observer in the penumbral 
region, C O or D 0, will see the moon partly projected on 
the sun, while if he chance to be situated at he will see 
an annular eclipse. To show how this is, we draw dotted 
lines from tangent to the moon. The angle between 
these lines represents the apparent diameter of the moon 
as seen from the earth. Continuing them to the sun, they 
show the apparent diameter of the moon as projected upon 
the sun. It will be seen that in the case supposed, when 

* It will be noted that all the figures of eclipses are necessarily drawn 
Very much out of proportion. Really the sun is 400 times the distance 
of the moon, which again is 60 times the radius of the earth. But it 
would be entirely impossible to draw a figure of this proportion ; we 
are therefore obliged to represent the earth as larger than the sun. and 
the moon as nearly half way between the earth and sun. 



ECLIPSES OF THE SUN. 



175 



the vertex of the shadow is between the earth and moon, 
the latter will necessarily appear smaller than the sun, and 
the observer will see a portion of the solar disk on all 
sides of the moon, as shown in Fig. 62. 

If the moon were a little nearer the earth than it is rep- 
resented in the figure, its shadow would reach the earth 




Fig. 62. — dark body op moon projected on sun during an 
annular eclipse. 

in the neighborhood of 0. We should then have a total 
eclipse at each point of the earth on which it fell. It will 
be seen, however, that a total or annular eclipse of the sun 
is visible only on a very small portion of the earth's sur- 
face, because the distance of the moon changes so little 
that the earth can never be far from the vertex 7^ of the 
shadow. As the moon moves around the earth from west 
to east, its shadow, whether the eclipse be total or annu- 
lar, moves in the same direction. The diameter of the 
shadow at the surface of the earth ranges from zero to 150 
miles. It therefore sweeps along a belt of the earth's sur- 
face of that breadth, in the same direction in which the 
earth is rotating. The velocity of the moon relative to 
the earth being 3400 kilometres per hour, the shadow 
would pass along with this velocity if the earth did not ro- 
tate, but owing to the earth's rotation the velocity relative 






176 ASTRONOMY. 

to points on its surface may range from 2000 to 3400 
kilometres (1200 to 2100 miles). 

The reader will readily understand that in order to see 
a total eclipse an observer must station himself before- 
hand at some point of the earth's surface over which the 
shadow is to pass. These points are generally calculated 
some years in advance, in the astronomical ephemerides, 
with as much precision as the tables of the celestial mo- 
tions admit of. 

It will be seen that a partial eclipse of the sun may be 
visible from a much larger portion of the earth's surface 
than a total or annular one. The space CD (Fig. 61) over 
which the penumbra extends is generally of about one half 
the diameter of the earth. Roughly speaking, a partial 
eclipse of the sun may sweep over a portion of the earth's 
surface ranging from zero to perhaps one fifth or one sixth 
of the whole. 

There are really more eclipses of the sun than of the 
moon. A year never passes without at least two of the 
former, and sometimes five or six, while there are rarely 
more than two eclipses of the moon, and in many years 
none at all. But at any one place more eclipses of the moon 
will be seen than of the sun. The reason of this is that 
an eclipse of the moon is visible over the entire hemi- 
sphere of the earth on which the moon is shining, and as it 
lasts several hours, observers who are not in this hemi- 
sphere at the beginning of the eclipse may, by the earth's ro- 
tation, be brought into it before it ends. Thus the eclipse 
will be seen over more than half the earth's surface. But, 
as we have just seen, each eclipse of the sun can be seen 
over only so small a fraction of the earth's surface as to 
more than compensate for the greater absolute frequency 
of solar eclipses. 

It will be seen that in order to have either a total or an- 
nular eclipse visible upon the earth, the line joining the 
centres of the sun and moon, being continued, must 
strike the earth. To an observer on this line, the centres 



RECURRENCE OF ECLIPSES. 177 

of the two bodies will seem to coincide. An eclipse in 
which this occurs is called a central one, whether it be 
total or annular. The accompanying figure will perhaps 
aid in giving a clear idea of the phenomena of eclipses of 
both sun and moon. 




Fig. 63. — comparison of shadow and penumbra op earth and 
moon. a is the position of the moon during a solar, b dur- 
ing a lunar eclipse. 

§ 4. THE RECURRENCE OP ECLIPSES. 

If the orbit of the moon around the earth were in or 
near the same plane with that of the latter around the sun 
— that is, in or near the plane of the ecliptic — it will be 
readily seen that there would be an eclipse of the sun at 
every new moon, and an eclipse of the moon at every 
full moon. But owing to the inclination of the moon's 
orbit, described in the last chapter, the shadow and 
penumbra of the moon commonly pass above or below the 
earth at the time of new moon, while the moon, at her 
full, commonly passes above or below the shadow of the 
earth. It is only when at the moment of new or full moon 
the moon is near its node that an eclipse can occur. 

The question now arises, how near must the moon be to 
its node in order that an eclipse may occur ? It is found 
by a trigonometrical computation that if, at the moment 
of new moon, the -moon is more than 18° -G from its 
node, no eclipse of the sun is possible, while if it is less 
than 13° -7 an eclipse is certain. Between these limits an 
eclipse may occur or fail according to the respective dis- 
tances of the sun and moon from the earth. Half way be- 
tween these limits, or say 16° from the node, it is an even 



178 ASTRONOMY. 

chance that an eclipse will occur ; toward the lower limit 
(13° • 1) the chances increase to certainty ; toward the 
upper one (18° • 6) they diminish to zero. The correspond- 
ing limits for an eclipse of the moon are 9° and 12£° — that 
is, if at the moment of full moon the distance of the 
moon from her node is greater than 12^° no eclipse can 
occur, while if the distance is less than 9° an eclipse is cer- 
tain. "We may put the mean limit at 11°. Since, in the 
long run, new and full moon will occur equally at all dis- 
tances from the node, there will be, on the average, sixteen 
eclipses of the sun to eleven of the moon, or nearlv fifty per 
cent more. 




Fig. 64.— Illustrating lunar eclipse at different distances from the node. The dark 
■circles are the earth's shadow, the centre of which is always in the ecliptic A B. The 
moon's orbit is represented by CD. At G the eclipse is central and total, at ^ it is 
partial, and at E there ia barely an eclipse. 

As an illustration of these computations, let us investigate the lim- 
its within which a central eclipse of the sun, total or annular, can 
occur. To allow of such an eclipse, it is evident, from an inspec- 
tion of Fig. 61 or 63 that the actual distance of the moon from 
the plane of the ecliptic must be less than the earth's radius, 
because the line joining the centres of the sun and earth always lies 
in this plane. This distance must, therefore, be less than 6370 kilo- 
metres. The mean distance of the moon being 384,000 kilometres, 
the sine of the latitude at this limit is ^ ||^y , and the latitude itself 
is 57'. The formula for the latitude is, by spherical trigonometry, 

sin latitude = sin i sin u, 

i being the inclination of the moon's orbit (5° 8'), and wthe distance 
of the moon from the node. The value of sin i is not far from -^, 
while, in a rough calculation, we may suppose the comparatively 
small angles u and the latitude to be the same as their sines. We 
may, therefore, suppose 

u = 11 latitude = 104°. 



RECURRENCE OF ECLIPSES. 179 

We therefore conclude that if, at the moment of new moon, the 
distance of the moon from the node is less than 10| ° there will be 
a central eclipse of the sun, and if greater than this there will not be 
such an eclipse. The eclipse limit may range half a degree or more 
on each side of this mean value, owing to the varying distance of 
the moon from the earth. Inside of 10" a central eclipse may be re- 
garded as certain, and outside of 11° as impossible. 



If the direction of the moon's nodes from the centre of 
the earth were invariable, eclipses conld occur only at the 
two opposite months of the year when the sun had nearly 
the same longitude as one node. For instance, if the lon- 
gitudes of the two opposite nodes were respectively 54° 
and 234°, then, since the sun must be within 12° of the 
node to allow of an eclipse of the moon, its longitude 
would have to be either between 42° and 66°, or between 
222° and 246°. But the sun is within the first of these re- 
gions only in the month of May, and within the second only 
during the month of November. Hence lunar eclipses 
<?ould then occur only during the months of May and No- 
vember, and the same would hold true of central eclipses 
of the sun. Small partial eclipses of the latter might be 
seen occasionally a day or two from the beginnings or ends 
of the above months, but they would be very small and 
quite rare. Now, the nodes of the moon's orbit were act- 
ually in the above directions in the year 1873. Hence 
during that year eclipses occurred only in May and No- 
vember. We may call these months the seasons of eclipses 
for 1873. 

But it was explained in the last chapter that there is a 
retrograde motion of the moon's nodes amounting to 19^° 
in a year. The nodes thus move back to meet the sun in 
its annual revolution, and this meeting occurs about 20 days 
earlier every year than it did the year before. The re- 
sult is that the season of eclipses is constantly shifting, so 
that each season ranges throughout the whole year in 18-6 
years. For instance, the season corresponding to that of 
November, 1873, had moved back to July and August in 






180 ASTRONOMY. 

1878, and will occur in May, 1882, while that of May, 
1873, will be shifting back to November in 1882. 

It may be interesting to illustrate this by giving the 
days in which the sun is in conjunction with the nodes of 
the moon's orbit during several years. 



Ascending Node. 

1879. January 24. 

1880. January 6. 
1880. December 18. 


Descending Node. 

1879. July 17. 

1880. June 27. 

1881. June 8. 


1881. November 30. 

1882. November 12. 

1883. October 25. 

1884. October 8. 


1882. May 20. 

1883. May 1. 

1884. April 12. 

1885. March 25, 



During these years, eclipses of the moon can occur only 
within 11 or 12 days of these dates, and eclipses of the 
sun only within 15 or 16 days. 

In consequence of the motion of the moon's node, three 
varying angles come into play in considering the occur- 
rence of an eclipse, the longitude of the node, that of the 
sun, and that of the moon. We may, however, simplify 
the matter by referring the directions of the sun and 
moon, not to any fixed line, but to the node — that is, we 
may count the longitudes of these bodies from the node 
instead of from the vernal equinox. We have seen in the 
last chapter that one revolution of the moon relatively to 
the node is accomplished, on the average, in 27-21222 
days. If we calculate the time required for the sun to re- 
turn to the node, we shall find it to be 346 • 6201 days. 

Now, let us suppose the sun and moon to start out 
together from a node. At the end of 346 • 6201 days the 
sun, having apparently performed nearly an entire rev- 
olution around the celestial sphere, will again be at the 
same node, which has moved back to meet it. But the 
moon will not be there. It will, during the interval, have 
passed the node 12 times, and the 13th passage will not 
occur for a week. The same thing will be true for 



RECURRENCE OF ECLIPSES. 181 

18 successive returns of the sun to the node ; we shall 
not find the moon there at the same time with the sun ; 
she will always have passed a little sooner or a little later. 
But at the 19th return of the sun and the 242d of the 
moon, the two bodies will be in conjunction within half 
a degree of the node. We find from the preceding 
periods that 

242 returns of the moon to the node require 6585 • 357 days. 
19 " " sun " " " 6585-780 " 

The two bodies will therefore pass the node within 10 
hours of each other. This conjunction of the sun and 
moon will be the 223d new moon after that from which 
we started. Now, one lunation (that is, the interval 
between two consecutive new moons) is, in the mean, 
29- 530588 days ; 223 lunations therefore require 6585-32 
days. The new moon, therefore, occurs a little before the 
bodies reach the node, the distance from the latter being 
that over which the moon moves in d -036, or the sun 
in d -459. We readily find this distance to be 28' of 
arc, somewhat less than the apparent diameter of either 
body. This would be the smallest distance from either 
node at which any new moon would occur during the 
whole period. The next nearest approaches would have 
occurred at the 35th and 47th lunations respectively. 
The 35th new moon would have occurred about 6° before 
the two bodies arrived at the node from which we started, 
and the 47th about l-J-° past the opposite node. No other 
new moon would occur so near a node before the 223d 
one, which, as we have just seen, would occur 0° 28' 
west of the node. This period of 223 new moons, or 18 
years 11 days, was called the /Saros by the ancient astron- 
omers. 

It will be seen that in the preceding calculations we have assumed 
the sun and moon to move uniformly, so that the successive new 
moon's occurred at equal intervals of 29-530588 days, and at equal 
angular distances around the ecliptic. In fact, however, the month- 
ly inequalities in the motion of the moon cause deviations from her 



182 ASTRONOMY. 

mean motion which amount to six degrees in either direction, while 
the annual inequality in the motion of the sun in longitude is nearly 
two degrees. Consequently, our conclusions respecting the point at 
which new moon occurs may be astray by eight degrees, owing to 
these inequalities. 

But there is a remarkable feature connected with the Saros which 
greatly reduces these inequalities. It is that this period of 6585^ 
days corresponds very nearly to an integral number of revolutions 
both of the earth round the sun, and of the lunar perigee around 
the earth. Hence the inequalities both of the moon and of the 
sun will be nearly the same at the beginning and the end of a Saros. 
In fact, 6585^- days is about 18 years and 11 days, in which time 
the earth will have made 18 revolutions, and about 11° on the 
19th revolution. The longitude of the sun will therefore be about 
11° greater than at the beginning of the period. Again, in the 
same period the moon's perigee will have made two revolutions, 
and will have advanced 13° 38' on the third revolution. The sun 
and moon being 11° further advanced in longitude, the conjunction 
will fall at the same distance from the lunar perigee within two or 
three degrees. Without going through the details of the calcula- 
tion, we may say as the result of this remarkable coincidence that 
the time of the 223d lunation will not generally be accelerated or 
retarded more than half an hour, though those of the intermediate 
lunations will sometimes deviate more than half a day. Also that 
the distance west of the node at which the new moon occurs will 
not generally differ from its mean value, 28' by more than 20'. 

In the preceding explanation, we have supposed the sun 
and moon to start out together from one of the nodes of 
the moon's orbit. It is evident, however, that we might 
have supposed them to start from any given distance east 
or west of the node, and should then at the end of the 223d 
lunation find them together again at nearly that distance 
from the node. For instance, on the 5th day of May, 
1864, at seven o'clock in the evening, Washington time, 
new moon occurred with the sun and moon 2° 25' west of 
the descending node of the moon's orbit. Counting for- 
ward 223 lunations, we arrive at the 16th day of May, 
1882, when we find the new moon to occur 3° 20' west of 
the same node. Since the character of the eclipse depends 
principally upon the relative position of the sun, the moon, 
and the node, the result to which we are led may be stated 
as follows : 

Let us note the time of the middle of any eclipse. 



RECURRENCE OF ECLIPSES, 183 

whether of the sun or of the moon. Then let us go for- 
ward 6585 days, 7 hours, 42 minutes, and we shall find 
another eclipse very similar to the first. Reduced to years, 
the interval will be 18 years and 10 or 11 days, according 
as a 29th day of February intervenes four or five times 
during the interval. This being true of every eclipse, it 
follows that if we record all the eclipses which occur dur- 
ing a period of 18 years, we shall find a new set to begin 
over again. If the period were an integral number of 
days, each eclipse of the new set would be visible in the 
same regions of the earth as the old one, but since there is 
a fraction of nearly 8 hours over the round number of 
days, the earth will be one third of a revolution further 
advanced before any eclipse of the new set begins. Each 
eclipse of the new set will therefore occur about one third 
of the way round the world, or 120° in longitude west of 
the region in which the old one occurred. The recur- 
rence will not take place near the same region until the end 
of three periods, or 54 years ; and then, since there is a 
slight deviation in the series, owing to each new or full 
moon occurring a little further west from the node, the 
fourth eclipse, though near the same region, will not 
necessarily be similar in all its particulars. For example, 
if it be a total eclipse of the sun, the path of the shadow 
may be a thousand miles distant from the j3ath of 54 years 
previously. 

As a recent example of the Saros, we may cite some 
total eclipses of the sun well known in reeeut times ; for 
instance : 

1842, July 8th, l h a.m., total eclipse observed in 
Europe ; 

1860, July 18th, 9 h a.m., total eclipse in America and 
Spain ; 

1878, July 29th, 4" p.m., one visible in Texas, Col- 
orado, and on the coast of Alaska. 

A yet more remarkable series of total eclipses of the 



184 ASTRONOMY. 

sun are those of the years 1850, 1868, 1886, etc., the dates 
and regions being : 

1850, August 7th, 4 h p.m., in the Pacific Ocean ; 

1868, August 17th, 12 h p.m., in India ; 

1886, August 29th, 8 h a.m., in the Central Atlantic 
Ocean and Southern Africa ; 

1901, September 9th, noon, in South America. 

This series is remarkable for the long duration of total- 
ity, amounting to some six minutes. 

Let us now consider a series of eclipses recurring at reg- 
ular intervals of 18 years and 11 days. Since every suc- 
cessive recurrence of such an eclipse throws the conjunc- 
tion 28' further toward the west of the node, the conjunc- 
tion must, in process of time, take place so far back from 
the node that no eclipse will occur, and the series will end. 
For the same reason there must be a commencement to 
the series, the first eclipse being east of the node. A new 
eclipse thus entering will at first be a very small one, but 
will be larger at every recurrence in each Saros. If it is 
an eclipse of the moon, it will be total from its 13th until 
its 36th recurrence. There will then be about 13 partial 
eclipses, each of which will be smaller than the last, when 
they will fail entirely, the conjunction taking place so far 
from the node that the moon does not touch the earth's 
shadow. The whole interval of time over which a series 
of lunar eclipses thus extend will be about 48 periods, or 
865 years. 

When a series of solar eclipses begins, the penumbra of 
the first will just graze the earth not far from one of the 
poles. There will then be, on the average, 11 or 12 partial 
eclipses of the sun, each larger than the preceding one, 
occurring at regular intervals of one Saros. Then the 
central line, whether it be that of a total or annular 
eclipse, will begin to touch the earth, and we shall have a 
series of 40 or 50 central eclipses. The central line will 
strike near one pole in the first part of the series ; in the 
equatorial regions about the middle of the series, and will 



CHARACTERS OF ECLIPSES. 185 

leave the earth by the other pole at the end. Ten or 
twelve partial eclipses will follow, and this particular se- 
ries will cease. The whole number in the series will aver- 
age between 60 and 70, occupying a few centuries over a 
thousand years. 

§ 5. CHARACTERS OF ECLIPSES. 

We have seen that the possibility of a total eclipse of the sun 
arises from the occasional very slight excess of the apparent angular 
diameter ol* the moon over that of the sun. This excess is so slight 
that such an eclipse can never last more than a few minutes. It 
may be of interest to point out the circumstances which favor a 
long duration of totality. These are : 

(1) That the moon should be as near as possible to the earth, or, 
technically speaking, in perigee, because its angular diameter as 
seen from the earth will then be greatest. 

(2) That the sun should be near its greatest, distance from the 
earth, or in apogee, because then its angular diameter will be the 
least. It is now in this position about the end of June ; hence the 
most favorable time for a total eclipse of very long duration is in 
the summer months. Since the moon must be in perigee and also 
between the earth and sun, it follows that the longitude of the 
perigee must be nearly that of the sun. The longitude of the sun 
at the end of June being 100°, this is the most favorable longi- 
tude of the moon's perigee. 

(3) The moon must be very near the node in order that the cen- 
tre of the shadow may fall near the equator. The reason of this con- 
dition is, that the duration of a total eclipse may be considerably 
increased by the rotation of the earth on its axis. We have seen 
that the shadow sweeps over the earth from west toward east with a 
velocity of about 3400 kilometres per hour. Since the earth rotates in 
the same direction, the velocity relative to the observer on the earth's 
surface will be diminished by a quantity depending on this velocity 
of rotation, and therefore greater, the greater the velocity. The 
velocity of rotation is greatest at the earth's equator, where it 
amounts to 1660 kilometres per hour, or nearly half the velocity of 
the moon's shadow. Hence the duration of a total eclipse may, with- 
in the tropics, be nearly doubled by the earth's rotation. When all 
the favorable circumstances combine in the way we have just de- 
scribed, the duration of a total eclipse within the tropics will be 
about seven minutes and a half. In our latitude the maximum du- 
ration will be somewhat less, or not far from six minutes, but it is. 
only on very rare occasions, hardly once in many centuries, that all 
these favorable conditions can be expected to concur. 

Of late years, solar eclipses have derived an increased in- 
terest from the fact that during the few minutes which. 



186 ASTRONOMY. 

they last they afford unique opportunities for investigating 
the matter which lies in the immediate neighborhood of 
the sun. Under ordinary circumstances, this matter is 
rendered entirely invisible by the effulgence of the solar 
rays which illuminate our atmosphere ; but when a body so 
distant as the moon is interposed between the observer and 
the sun, the rays of the latter are cut off from a region a 
hundred miles or more in extent. Thus an amount of 
darkness in the air is secured which is impossible under 
any other circumstances when the sun is far above the 
horizon. Still this darkness is by no means complete, because 
the sunlight is reflected from the region on which the sun 
is shining. An idea of the amount of darkness may be 
gained by considering that the face of a watch can be read 
during an eclipse if the observer is careful to shade his 
eyes from the direct sunlight during the few minutes be- 
fore the sun is entirely covered ; that stars of the first 
magnitude can be seen if one knows where to look for 
them ; and that all the prominent features of the land- 
scape remain plainly visible. An account of the investi- 
gations made during solar eclipses belongs to the physical 
constitution of the sun, and will therefore be given in a 
subsequent chapter. 

Oceultation of Stars by the Moon. — A phenomenon 
which, geometrically considered, is analogous to an eclipse 
of the sun is the oceultation of a star by the moon. 
Since all the bodies of the solar system are nearer than the 
fixed stars, it is evident that they must from time to time 
pass between us and the stars. The planets are, however, 
so small that such a passage is of very rare occurrence, 
and when it does happen the star is generally so faint 
that it is rendered invisible by the superior light of the 
planet before the latter touches it. There are not more 
than one or two instances recorded in astronomy of a well- 
authenticated observation of an actual oceultation of a star 
by the opaque body of a planet, although there are several 
cases in which a planet has been known to pass over a star. 



OCCULTATION OF STABS. 187 

But the moon is so large and her angular motion so rapid, 
that she passes over some star visible to the naked eye 
every few days. Such phenomena are termed occultations 
of stars by the moon. It must not, however, be supposed 
that they can be observed by the naked eye. In general, 
the moon is so bright that only stars of the first magnitude 
can be seen in actual contact with her limb, and even then 
the contact must be with the unilluminated limb. But 
with the aid of a telescope, and the predictions given in 
the Ephemeris, two or three of these occultations can be 
observed during nearly every lunation. 






CHAPTER VIII. 

THE EAETH. 

Our object in the present chapter is to trace the effects 
of terrestrial gravitation and to study the changes to 
which it is subject in various places. Since every part 
of the earth attracts every other part as well as every 
object upon its surface, it follows that the earth and 
all the objects that we consider terrestrial form a sort 
of system by themselves, the parts of which are firmly 
bound together by their mutual attraction. This attrac- 
tion is so strong that it is found impossible to project 
any object from the surface of the earth into the celestial 
spaces. Every particle of matter now belonging to the 
earth must, so far as we can see, remain upon it forever. 

§ 1. MASS AND DENSITY OF THE EARTH. 

We begin by some definitions and some principles re- 
specting attraction, masses, weight, etc. 

The mass of a body may be defined as the quantity of 
?natter which it contains. 

There are two ways to measure this quantity of mat- 
ter : (1) By the attraction or weight of the body — this 
weight being, in fact, the mutual force of attraction be- 
tween the body and the earth ; (2) By the inertia of the 
body, or the amount of force which we must apply to it in 
order to make it move with a definite velocity. Mathe- 
matically, there is no reason why these two methods should 
give the same result, but by experiment it is found that 



MASS OF THE EARTH. 189 

the attraction of all bodies is proportional to their inertia. 
In other words, all bodies, whatever their chemical consti- 
tution, fall exactly the same number of feet in one second 
under the influence of gravity, supposing them in a vacu- 
um and at the same place on the earth's surface. Although 
the mass of a body is most conveniently determined by its 
weight, yet mass and weight must not be confounded. 

The weight of a body is the apparent force with which 
it is attracted toward the centre of the earth. As we 
shall see hereafter, this force is not the same in all parts of 
the earth, nor at different heights above the earth's sur- 
face. It is therefore a variable quantity, depending upon 
the position of the body, while the mass of the body is re- 
garded as something inherent in it, which remains constant 
wherever the body may be taken, even if it is carried 
through the celestial spaces, where its weight would be 
reduced to almost nothing. 

The unit of mass which we may adopt is arbitrary ; in 
fact, in different cases different units will be more con- 
venient. Generally the most convenient unit is the weight 
of a body at some fixed place on the earth's surface — the 
city of Washington, for example. Suppose we take such 
a portion of the earth as will weigh one kilogram in Wash- 
ington, we may then consider the mass of that particular 
lot of earth or rock as a kilogram, no matter to what part 
of the universe we take it. Suppose also that we could 
bring all the matter composing the earth to the city of 
Washington, one kilogram at a time, for the purpose of 
weighing it, returning each kilogram to its place in the 
earth immediately after weighing, so that there should be 
no disturbance of the earth itself. The sum total of the 
weights thus found would be the mass of the earth, and 
would be a perfectly definite quantity, admitting of being 
expressed in kilograms or pounds. We can readily cal- 
culate the mass of a volume of water equal to that of the 
earth because we know the magnitude of the earth in 
litres, and the mass of one litre of water. Dividing this 



190 ASTRONOMY. 

into the mass of the earth, supposing ourselves able to de- 
termine this mass, and we shall have the specific gravity, 
or what is more properly called the density of the earth. 

What we have supposed for the earth we may imagine 
for any heavenly body — namely, that it is brought to the 
city of Washington in small pieces, and there weighed one 
piece at a time. Thus the total mass of the earth or any 
heavenly body is a perfectly defined and determined 
quantity. 

It may be remarked in this connection that our units o! 
weight, the pound, the kilogram, etc., are practically unita 
of mass rather than of weight. If we should weigh out 
a pound of tea in the latitude of Washington, and then 
take it to the equator, it would really be less heavy at the 
equator than in Washington ; but if we take a pound 
weight with us, that also would be lighter at the equator, 
so that the two would still balance each other, and the tea 
would be still considered as weighing one pound. Since 
things are actually weighed in this way by weights which 
weigh one unit at some definite place, say Washington, 
and which are carried all over the world without being 
changed, it follows that a body which has any given 
weight in one place will, as measured in this way, have 
the same apparent weight in any other place, although its 
real weight will vary. But if a sj3ring balance or any 
other instrument for determining actual weights were 
adopted, then we should find that the weight of the same 
body varied as we took it from one part of the earth to 
another. Since, however, we do not use this sort of an 
instrument in weighing, but pieces of metal which are 
carried about without change, it follows that what we call 
units of weight are properly units of mass. 

Density of the Earth. — We see that all bodies around 
us tend to fall toward the centre of the earth. According 
to the law of gravitation, this tendency is not simply a 
single force directed toward the centre of the earth, but 
is the resultant of an infinity of separate forces arising from 



MASS OF THE EARTH. 191 

the attractions of all the separate parts which compose the 
earth. The question may arise, how do we know that each 
particle of the earth attracts a stone which falls, and that 
the whole attraction does not reside in the centre ? The 
proofs of this are numerous, and consist rather in the 
exactitude with which the theory represents a great mass 
of disconnected phenomena than in any one principle ad- 
mitting of demonstration. Perhaps, however, the most 
conclusive proof is found in the observed fact that masses 
of matter at the surface of the earth do really attract each 
other as required by the law of Newton. It is found, for 
example, that isolated mountains attract a plumb-line in 
their neighborhood. The celebrated experiment of Cav- 
endish was devised for the purpose of measuring the at- 
traction of globes of lead. The object of measuring this 
attraction, however, was not to prove that gravitation re- 
sided in the smallest masses of matter, because there was 
no doubt of that, but to determine the mean density of the 
earth, from which its total mass may be derived by simply 
multiplying the density by the volume. 

It is noteworthy that though astronomy affords us the 
means of determining with great precision the relative 
masses of the earth, the moon, and all the planets, it does 
not enable us to determine the absolute mass of any hea- 
venly body in units of the weights we use on the earth. 
We know, for instance, from astronomical research, that 
the sun has about 328,000 times the mass of the earth, 
and the moon only -fa of this mass, but to know the abso- 
lute mass of either of them we must know how many 
kilograms of matter the earth contains. To determine 
this, we must know the mean density of the earth, and this 
is something about which direct observation can give us no 
information, because we cannot penetrate more than an 
insignificant distance into the earth's interior. The only 
way to determine the density of the earth is to find how 
much matter it must contain in order to attract bodies on 
its surface with a force equalto their observed weight — 






192 ASTRONOMY. 

that is, with such intensity that at the equator a body shall. 
fall nearly five metres in one second. To find this we- 
must know the relation between the mass of a body and 
its attractive force. This relation can be found only by 
measuring the attraction of a body of known mass. An 
attempt to do this was made by Maskelyne, Astronomer 
Royal of England, toward the close of the last century, 
the attracting object he selected being Mount Schehallien 
in Scotland. The specific gravity of the rocks composing 
this mountain was well enough known to give at least an 
approximate result. The density of the earth thus found 
was 4*71. That is, the earth has 4.71 times the mass of 
an equal volume of water. This result is, however, un- 
certain, owing to the necessary uncertainty respecting the 
density of the mountain and the rocks below it. 

The Cavendish experiment for determining the attrac- 
tion of a pair of massive balls affords a much more perfect 
method of determining this important element. The 
most careful experiments by this method were made by 
Baily of England about the year 1845. The essential 
parts of the apparatus which he used are as follows : 

A long narrow table T bears two massive spheres of lead. 
W W, one at each end. This table admits of being 
turned around on a pivot in a horizontal direction. 
Above it is suspended a balance — that is, a very light deal 
rod e with a weight at each end suspended horizontally 
by a fine silver wire or fibre of silk F E. The weights to- 
be attracted are attached to each end of the deal rod. The 
right-hand one is visible, while the other is hidden be- 
hind the left-hand weight W. In this position it will be 
seen that the attraction of the weights W tends to turn 
the balance in a direction opposite that of the hands of a 
watch. The fact is, the balance begins to turn in this di- 
rection, and being carried by its own momentum beyond 
the point of equilibrium, comes to rest by a twist of the 
thread. It is then carried part of the way back to its- 
original position, and thus makes several vibrations which 



DENSITY OF THE EARTH. 



193 



require several minutes. At length it comes to rest in a 
position somewhat different from its original one. This 
position and the times of vibration are all carefully noted. 
Then the table T is turned nearly end for end, so that one 
weight W shall be between the observer and the right- 
hand ball, while the other weight is beyond the left-hand 
ball, and the observation is repeated. A series of observa- 
tions made in this way include attractions in alternate di- 




Fra. 65. 



Tections, giving a result from which accidental errors will 
be very nearly eliminated. 

A third method of determining the density of the earth 
is founded on observations of the change in the intensity 
of gravity as we descend below the surface into deep 
mines. The principles on which this method rests will be 
explained presently. The most careful application of it 
was made by Professor Airy in the Harton Colliery, Eng- 



194 ASTRONOMY. 

land. The results of this and the other methods are as 
follows : 

Cavendish and Hutton, from the attraction of balls, 5 • 32 
Keich, " " " 5-58 

Baily, " " " 5-66 

Maskelyne, from the attraction of Schehallien 4*71 

Airy, from gravity in the Harton Colliery 6-56 

Of these different results, that of Baily is probably the 
best, and the most probable mean density of the earth is 
about 5f times that of water. This is more than double 
the mean specific gravity of the materials which compose 
the surface of the earth ; it follows, therefore, that the in- 
ner portions of the earth are much more dense than its 
outer portions. 

§ 2. LAWS OF TERRESTRIAL GRAVITATION. 

The earth being very nearly spherical, certain theorems 
respecting the attraction of spheres may be applied to it. 
The fundamental theorems may be regarded as those 
which give the attraction of a spherical shell of matter. 
The demonstration of these theorems requires the use of 
the Integral Calculus, and will be omitted here, only the 
conditions and the results being stated. Let us then im- 
agine a hollow shell of matter, of which the internal and 
external surfaces are both spheres, attracting any other 
masses of matter, a small particle we may suppose. This 
particle will be attracted by every particle of the shell 
with a force inversely as the square of its distance from it. 
The total attraction of the shell will be the resultant of 
this infinity of separate attractive forces. Determining 
this resultant by the Integral Calculus, it is found that : 

Theorem I. — If the particle he outside the shell, it will 
he attracted as if the whole mass of the shell were con- 
centrated in its centre. 

Theorem II. — If the particle he inside the shell, the op- 



ATTRACTION OF SPHERES. 195 

posite attractions in every direction will neutralize each 
other, no matter whereabouts in the interior the particle 
may be, and the resultant attraction of the shell will there- 
fore be zerc. 

To apply this to the attraction of a solid sphere, let us 
first suppose a body either outside the sphere or on its sur- 
face. If we conceive the sphere as made up of a great 
number of spherical shells, the attracted point will be ex- 
ternal to all of them. Since each shell attracts as if its 
whole mass were in the centre, it 
follows that the whole sphere at- 
tracts a body upon the outside of 
its surface as if its entire mass 
were concentrated at its centre. 

Let us now suppose the attract- 
ed particle inside the sphere, as 
at P, Fig. 66, and imagine a 
spherical surface P Q concentric 
with the sphere and passing 
through the attracted particle. FlG * 66, 

All that portion of the sphere lying outside this spherical 
surface will be a spherical shell having the particle inside 
of it, and will therefore exert no attraction whatever on 
the particle. That portion inside the surface will con- 
stitute a sphere with the particle on its surface, and will 
therefore attract as if all this portion were concentrated 
in the centre. To find what this attraction will be, let us 
first suppose the whole sphere of equal density. Let us 
put 

a, the radius of the entire sphere. 

r, the distance P C of the particle from the centre. 
The total volume of matter inside the sphere P Q will 

then be, by geometry, - n r\ Dividing by the square of 

the distance r, we see that the attraction will be repre- 
sented by 

•i 
3*r; 




196 ASTRONOMY. 

that is, inside the sphere the attraction will be directly as 
the distance of the particle from the centre. If the par- 
ticle is at the surface we have r = a, and the attraction is 

4 

4 
Outside the surface the whole volume of the sphere -5 n a? 

o 

will attract the particle, and the attraction will be 

4 a 3 

— IT . 

Jf we put r = a in this formula, we shall have the same 
result as before for the surface attraction. 

Let us next suppose that the density of the sphere va- 
ries from its centre to its surface, but in such a way as to 
be equal at equal distances from the centre. We may 
then conceive of it as formed of an infinity of concentric 
spherical shells, each homogeneous in density, but not of 
the same density with the others. Theorems I. and II. 
will then still apply, but their result will not be the same 
as in the case of a homogeneous sphere for a particle in- 
side the sphere. Referring to Fig. 66, let us put 

D, the mean density of the shell outside the particle P, 
D', the mean density of the portion P Q inside of P. 
We shall then have : 

4 
Volume of the shell, — n (a* — r 3 ). 

o 

4 

Vol ume of the inner sphere, — n r*. 

o 

4 
Mass of the shell = vol. x D = q- n D (V — r*). 

4 
Mass of the inner sphere = vol. x D' = x- n D' r\ 

Mass of whole sphere = sum of masses of shell and inner 

sphere = | n [p a 3 + (&' - D) r^. 



ATTRACTION OF SPHERES. 19? 

Attraction of the whole sphere upon a point at its sur- 
Mass 4 / _ x r 3 \ 

Attraction of tlie inner sphere upon a point at 
Mass 4 

If, as in the case of the earth, the density continually in- 
creases toward the centre, the value of D' will increase 
also as r diminishes, so that gravity will diminish less 
rapidly than in the case of a homogeneous sphere, and 
may, in fact, actually increase. To show this, let us sub- 
tract the attraction at P from that at the surface. The 
difference will give : 

Diminution at P = t n (z> a + (B' - D) T — - D' r) . 

Now, let us suppose r a very little less than #, and put 

r — a — d, 

d will then be the depth of the particle below the surface. 
Cubing this value of r, neglecting the higher powers of 
d, and dividing by a% we find, 

r % 

—^ — a — Zd. 

a 

Substituting in the above equation, the diminution of grav- 
ity at P becomes 

~7t(ZP- 2J)')d. 

We see that if SD < 2Z>', that is, if the density at the 
surface is less than f of the mean density of the whole in- 
ner mass, this quantity will become negative, showing that 
the force of gravity will be less at the surface than at a 
small depth in the interior. But it must ultimately 
diminish, because it is necessarily zero at the centre. 
It was on this principle that Professor Airy determined 
the density of the earth by comparing the vibrations 



198 ASTRONOMY. 

of a pendulum at the bottom of the Harton Colliery, and 
at the surface of the earth in the neighborhood. At the 
bottom of the mine the pendulum gained about 2 s • 5 per 
day, showing the force of gravity to be greater than at the 
surface. 



§ 3. FIGURE AND MAGNITUDE OF THE EARTH. 

If the earth were fluid and did not rotate on its axis, it 
would assume the form of a perfect sphere. The opinion 
is entertained that the earth was once in a molten state, 
and that this is the origin of its present nearly spherical 
form. If we give such a sphere a rotation upon its axis, 
the centrifugal force at the equator acts in a direction op- 
posed to gravity, and thus tends to enlarge the circle of 
the equator. It is found by mathematical analysis that 
the form of such a revolving fluid sphere, supposing it to 
be perfectly homogeneous, will be an oblate ellipsoid — that 
is, all the meridians will be equal and similar ellipses, hav- 
ing their major axes in the equator of the sphere and their 
minor axes coincident with the axis of rotation. Our earth, 
however, is not wholly fluid, and the solidity of its conti- 
nents prevents its assuming the form it would take if the 
ocean covered its entire surface. When we speak of the fig- 
ure of the earth hereafter, we mean, not the outline of the 
solid and liquid portions respectively, but the figure which 
it would assume if its entire surface were an ocean. Let us 
imagine canals dug down to the ocean level in every direc- 
tion through the continents, and the water of the ocean 
to be admitted into them. Then the curved surface touch- 
ing the water in all these canals, and coincident with the 
surface of the ocean, is that of the ideal earth considered 
by astronomers. By the figure of the earth is meant the 
figure of this liquid surface, without reference to the in- 
equalities of the solid surface. 

We cannot say that this ideal earth is a perfect ellipsoid, 
because we know that the interior is not homogeneous. 



MEASUREMENT OF TEE EAETE. 



199 



but all the geodetic measures heretofore made are so nearly 
represented by the hypothesis of an ellipsoid that the lat- 
ter is considered as a very close approximation to the true 
tigure. The deviations hitherto noticed are of so irregu- 
lar a character that they have not yet been reduced to any 
certain law. The largest which have been observed seem 
to be due to the attraction of mountains, or to inequalities 
of density beneath the surface. 

Method of Triangulation. — Since it is practically im- 
possible to measure around or through the earth, the mag- 
nitude as well as the form of our planet has to be found 
by combining measurements on its surface with astronom- 
ical observations. Even a measurement on the earth's 
surface made in the usual way of surveyors would be im- 
practicable, owing to the intervention of mountains, rivers, 
forests, and other natural obstacles. The method of tri- 
angulation is therefore universally adopted for measure- 
ments extending over large areas. A triangulation is ex- 
ecuted in the following way : Two points, a and b, a few 




Fig. 67. — a part op the French triangulation near pares. 

miles apart, are selected as the extremities of a base-line. 
They must be so chosen that their distance apart can be 
accurately measured by rods ; the intervening ground 
should therefore be as level and free from obstruction as 
possible. One or more elevated points, EF, etc., must 
be visible from one or both ends of the base-line. By 



200 ASTRONOMY. 

means of a theodolite and by observation of the pole-star, 
the directions of these points relative to the meridian are 
accurately observed from each end of the base, as is also 
the direction a b of the base-line itself. Suppose F to 
be a point visible from each end of the base, then in the 
triangle abFvre have the length a b determined by actual 
measurement, and the angles at a and b determined by ob- 
servations. With these data the lengths of the sides a F 
and bFsiTG determined by a simple trigonometrical com- 
putation. 

The observer then transports his instruments to F, and 
determines in succession the direction of the elevated 
points or hills D E G HJ, etc. He next goes in succes- 
sion to each of these hills, and determines the direction of 
all the others which are visible from it. Thus a network 
of triangles is formed, of which all the angles are observed 
with the theodolite, while the sides are successively calcu- 
lated trigonometrically from the first base. For instance, 
we have just shown how the side aF is calculated ; this 
forms a base for the triangle EFa, the two remaining 
sides of which are computed. The side EF forms the 
base of the triangle G EF, the sides of which are calcu- 
lated, etc. In this operation more angles are observed 
than are theoretically necessary to calculate the triangles. 
This surplus of data serves to insure the detection of any 
errors in the measures, and to test their accuracy by the 
agreement of their results. Accumulating errors are fur- 
ther guarded against by measuring additional sides from 
time to time as opportunity offers. 

Chains of triangles have thus been measured in Russia 
from the Danube to the Arctic Ocean, in England and 
France from the Hebrides to Algiers, in this country down 
nearly our entire Atlantic coast and along the great lakes, 
and through shorter distances in many other countries. 
An east and west line is now being run by the Coast Sur- 
vey from the Atlantic to the Pacific Ocean. Indeed it 
may be expected that a network of triangles will be grad- 



MAGNITUDE OF THE EARTH. 201 

ually extended over the surface of every civilized country, 
in order to construct perfect maps of it. 

Suppose that we take two stations situated north 
.and south of each other, determine the latitude of each, 
and measure the distance between them. It is evident that 
by dividing the distance in kilometres by the difference of 
latitude in degrees, we shall have the length of one degree 
of latitude. Then if the earth were a sphere, we should 
at once have its circumference by multiplying the length 
of one degree by 360. It is thus found, in a rough way, 
that the length of a degree is a little more than 111 kilo- 
metres, or between 69 and 70 English statute miles. Its 
•circumference is therefore about 40,000 kilometres, and 
its diameter between 12,000 and 13,000.* 

Owing to the ellipticity of the earth, the lengtl of one 
degree varies with the latitude and the direction in which 
it is measured. The next step in the order of accuracy is 
i:o find the magnitude and the form of the earth from 
measures of long arcs of latitude (and sometimes of longi- 
tude) made in different regions, especially near the equa- 
tor and in high latitudes. But we shall still find that dif- 
ferent combinations of measures give slightly different re- 
sults, both for the magnitude and the ellipticity, owing 
to the irregularities in the direction of attraction which we 
lave already described. The problem is therefore to find 
what ellipsoid will satisfy the measures with the least sum 
total of error. New and more accurate solutions will be 
reached from time to time as geodetic measures are extend- 
ed over a wider area. The following are among the most 
recent results hitherto reached : Listing- of Gottino-en 
in 1878 found the earth's polar semidiameter,6355 • 270 kilo* 

* When the metric system was originally designed by the French, it 
was intended that the kilometre should be ToiWo of the distance from 
the pole of the earth to the equator. This would make a degree of the 
meridian equal, on the average, to lll-£ kilometres. But, owiug to the 
practical difficulties of measuring a meridian of the earth, the corre- 
spondence with the metre actual!)- adopted is not exact. 



202 



ASTRONOMY. 



metres; earth's equatorial semidiameter, 6377-377 kilo- 
metres ; earth's compression, 28 g 5 of the equatorial di- 
ameter ; earth's eccentricity of meridian, 0-08319. An- 
other result is that of Captain Clarke of England, who 
found : Polar semidiameter, 6356-456 * kilometres \ equa- 
torial semidiameter, 6378-191 kilometres. 

It was once supposed that the measures were slightly bet- 
ter represented by supposing the earth to be an ellipsoid 
with three unequal axes, the equator itself being an ellipse 
of which, the longest diameter was 500 metres, or about 
pne third of a mile, longer than the shortest. This result 
was probably due to irregularities of gravity in those parts 
of the continents over which the geodetic measures have 
extended and is now abandoned. 

Geographic and Geocentric Latitudes. — An obvious re- 
sult of the ellipticity of the earth is that the plumb-line 




does not point toward the earth's centre. Let Fig. 68 
represent a meridional section of the earth, JV S being the 
axis of rotation, E Q the plane of the equator, and O the 
position of the observer. The line H R, tangent to the 

* Captain Clarke's results are given in feet, the polar radius being 
20,854,895 feet. In changing to metres, the logarithm of the factor has. 
been taken as 9.4840071. 



FORCE OF GRAVITY. 203 

earth at 0, will then represent the horizon of the observer, 
while the line Z iV 7 , perpendicular to II R, and therefore 
normal to the earth at (9, will be vertical as determined 
by the plumb-line. The angle N' Q, or Z Q\ which 
the observer's zenith makes with the equator, will then be 
his astronomical or geographical latitude. This is the lat- 
itude which in practice we nearly always have to use, be- 
cause we are obliged to determine latitude by astronomical 
observation, and not by measurement from the equator. 
We cannot determine the direction of the true centre C of 
the earth by direct observation of any kind, but only that 
of the plumb-line, or of the perpendicular to a fluid sur- 
face. Z Q' is therefore the astronomical latitude. If, 
however, we conceive the line C Oz drawn from the cen- 
tre of the earth through 0, z will be the observer's geo- 
centric zenith, while the angle G Q will be his geocen- 
tric latitude. It will be observed that it is the geocentric 
and not the geographic latitude which gives the true posi- 
tion of the observer relative to the earth's centre. The 
difference between the two latitudes is the anode CO J¥ f 
or Z z ; this is called the angle of the vertical. It is zero 
at the poles and at the equator, because here the normals 
pass through the centre of the ellipse, and it attains its 
maximum of 11' 30" at latitude 45°. It will be seen that 
the geocentric latitude is always less than the geographic. 
In north latitudes the geocentric zenith is south of the ap- 
parent zenith and in southern latitudes north of it, being 
nearer the equator in each case. 



§ 4. CHANGE OF GRAVITY WITH THE LATI- 
TUDE. 

If the earth were a homogeneous sphere, and did not rotate on 
•its axis, the intensity of gravity would be the same over its entire 
surface. There is a slight variation from two causes, namely, (1) 
The elliptic form of our globe, and (2) the centrifugal force gene- 
rated by its rotation on its axis. Strictly speaking, tlie latter is 
not a change in the real force of gravity, or of the earth's attrac- 
tion, but only an apparent force of another kind acting in oppo- 
sition to gravity. 



204: ASTRONOMY. 

The intensity of gravity is measured by the velocity which a 
heavy body in a vacuum will acquire in a unit of time, say one second. 
Either 10 metres or 82 feet may be regarded as a rough approxima- 
tion to its value. There are, however, so many practical difficul- 
ties in the way of measuring with precision the distance a body 
falls in one second, that the force of gravity is, in practice, deter- 
mined indirectly by finding the length of the second's pendulum. 
It is shown in mechanics that if a pendulum of length L vibrates 
in a time T, a heavy body will in this time T fall through the 
space 7T 2 L, n being the ratio of the circumference of a circle to its 
diameter. (tt=3. 14159 . . . tt 2 = 9- 869604.) Therefore, to find the 
force of gravity we have only to determine the length of the 
second's pendulum, and multiply it by this factor. 

The determination of the mean attractive force of the earth is 
important in order that we may compute its action on the moon 
and other heavenly bodies, while the variations of this attraction 
afford us data for judging of the variations of density in the earth's 
interior. Scientific expeditions have therefore taken pains to 
determine the length of the second's pendulum at numerous points 
on the globe. To do this, it is not necessary that they should 
actually measure the length of the pendulum at all the places they 
visit. They have only to carry some one pendulum of a very solid 
construction to each point of observation, and observe how many 
vibrations it makes in a day. They know that the force of gravity 
is proportional to the square of the number of vibrations. Before 
and after the voyage, they count the vibrations at some standard 
point — London for instance. Thus, by simply squaring the number 
of vibrations and comparing the squares, they have the ratio 
which gravity at various points of the earth's surface bears to 
gravity at London. It is then only necessary to determine the 
absolute intensity of gravity at London to infer it at all the 
other points for which the ratio is known. From a great number 
of observations of this kind, it is found that the length of the 
second's pendulum in latitude </> may be nearlv represented by the 
equation, 

L = m • 99099 (1 + • 00520 sin 2 <fi). 

From this, the force of gravity is found by multiplying by 
7T 2 = 9-8696, giving the result : 

g> = 9 m • 7807 (1 + • 00520 sin 2 <p). 

These formulae show that the apparent force of gravity increases 
by a little more than -^5 °f * ts whole amount from the equator to 
the poles. We can readily calculate how much of the diminution* 
at the equator is due to the centrifugal force of the earth's rotation. 
By the formulas of mechanics, the centrifugal force is given by the 
equation, 

f=—*- r 



TERRESTRIAL GRAVITY. 205 

T being the time of one revolution, and r the radius of the circle of 
rotation. Supposing the earth a sphere, which will cause no 
important error in our present calculation, the distance of a point 
on the earth's surface in latitude from the axis of rotation of the 
earth is, 

r — a cos </>, 

a being the earth's radius. The centrifugal force in latitude is 
therefore 

- _ 4 7T 2 a cos <p 

But this force does not act in the direction normal to the earth's 
surface, but perpendicular to the axis of the earth, which direction 
makes the angle with the normal. We may therefore resolve the 
force into two components, one, f sin 0, along the earth's surface 
toward the equator, the other,/ cos 0, downward toward its centre. 
The first component makes the earth an oblate ellipsoid, as already 
shown, while the second acts in opposition to gravity. The cen- 
trifugal force, therefore, diminishes gravity by the amount, 

- 4 7r 2 a cos 2 6 
/cos = — 1. 

T, the sidereal day, is 86,164 seconds of mean time, while «, for 
the equator, is 6,377,377 metres. Substituting in this expression, 
the centrifugal force becomes 

/cos <p = m • 03391 cos 2 = m • 03391 (1 — sin 2 0), 

or at the equator a little more than -^ the force of gravity. The 
expression for the apparent force of gravity given by observation, 
which we have already found, may be put in the form, 

g' = 9 m • 7807 + m • 05087 sin 2 0. 

This is the true force of gravity diminished by the centrifugal 
force ; therefore, to find that true force we must add the centri- 
fugal force to it, giving the result : 

g = 9 m • 8146 + m • 01696 sin 3 <f> 
= 9 m -8146 (1 + 0-001728 sin 2 0), 

for the real attraction of the spheroidal earth upon a body on its 
surface in latitude <j>. 

It will be interesting to compare this result with the attraction 
of a spheroid having the same ellipticity as the earth. It is found 
by integration that if e, supposed small, be the eccentricity of a 
homogeneous oblate ellipsoid, and g its attraction upon a body 
on its equator, its attraction at latitude will be given by the 
equation, 

9 = go (I + ^- o -sm 2 0> 



206 ASTRONOMY. 

In the case of the earth, e = 0-0817 ; T V 2 =0-000067 ; so that 
the expression for gravity would be, 

g =g (1 + 0- 000607 sin 2 £). 

"We see that the factor of sin 2 <p, which expresses the ratio in 
which gravity at the poles exceeds that at the equator, has less than 
half the value (-001780;, which we have found from observation. 
This difference arises from the fact that the earth is not homogene- 
ous, but increases in density from the surface toward the centre. 
To see how this result follows, let us first inquire how the earth 
would attract bodies where its surface now is if its whole mass 
were concentrated in its centre. The distance of the equator 
from the centre is to that of the poles from the centre as 1 to 
V\ — e' 2 . Therefore, in the case supposed, attraction at the equator 
would be to attraction ut the poles as 1 — e 1 to 1. The ratio of in- 
crease of attraction at the poles is therefore in this extreme case 
about ten times what it is for the homogeneous ellipsoid. We con- 
clude, therefore, that the more nearly the earth approaches this 
extreme case — that is, the more it increases in density toward the 
centre — the greater will be the difference of attraction at the poles 
and the equator. 



§ 5. MOTION OF THE EARTH'S AXIS, OR PRE- 
CESSION OF THE EQUINOXES. 

Sidereal and Equinoctial Year. — In describing the ap- 
parent motion of the sun, two ways were shown of find- 
ing the time of its apparent revolution around the sphere 
— in other words, of fixing the length of a year. One of 
these methods consists in finding the interval between suc- 
cessive passages through the equinoxes, or, which is the 
same thing, across the plane of the equator, and the other 
by finding when it returns to the same position among 
the stars. Two thousand years ago, Hippaechus found, 
by comparing his own observations with those made two 
centuries before by Timochaeis, that these two methods 
of fixing the length of the year did not give the same 
result. It had previously been considered that the length 
of a year was about 365J days, and in attempting to correct 
this period by comparing his observed times of the sun's 
passing the equinox with those of Timochaeis, Hippae- 
chus found that it required a diminution of seven or eight 



LENGTH OF THE TEAR. 207 

minutes. He therefore concluded that the true length of 
the equinoctial year was 365 days, 5 hours, and about 53 
minutes. When, however, he considered the return, not 
to the equinox, but to the same position relative to the 
bright star Spica Virginis, he found that it took some 
minutes more than 365J days to complete the revolution. 
Thus there are two years to be distinguished, the tropical 
or equinoctial year and the sidereal year. The first is 
measured by the time of the earth's return to the equinox ; 
the second by its return to the same position relative to the 
stars. Although the sidereal year is the correct astronom- 
ical period of one revolution of the earth around the sun, 
yet the equinoctial year is the one to be used in civil life, 
because it is upon that year that the change of seasons 
depends. Modern determinations show the respective 
Wg+-hs of the two years to be : 

Sidereal year, 365 d 6 h 9 ra 9 s = 365 d . 25636. 
Equinoctial year, 365 d 5 h 48 m 46 s = 365 d - 24220. 

It is evident from this difference between the two years 
that the position of the equinox among the stars must be 
changing, and must move toward the west, because the 
equinoctial year is the shorter. This motion is called the 
precession of the equinoxes, and amounts to about 50" 
per year. The equinox being simply the point in which 
the equator and the ecliptic intersect, it is evident that it 
can change only through a change in one or both of these 
circles- Hipparchus found that the change was in the 
equator, and not in the ecliptic, because the declinations of 
the stars changed, while their latitudes did not.* Since 

* To describe the theory of the ancient astronomers with perfect 
correctness, we ought to say that they considered the planes both of the 
equator and ecliptic to be invariable and the motion of precession to 
be due to a slow revolution of the whole celestial sphere around the 
pole of the ecliptic as an axis. This would produce a change in the 
position of the stars relative to the equator, but not relative to the 
ecliptic. 



208 ASTRONOMY. 

the equator is defined as a circle everywhere 90° distant 
from the pole, and since it is moving among the stars, it 
follows that the pole must also be moving among the stars. 
But the pole is nothing more than the point in which the 
earth's axis of rotation intersects the celestial sphere : it 
must be remembered too that the position of this pole in 
the celestial sphere depends solely upon the direction of 
the earth's axis, and is not changed by the motion of the 
earth around the sun, because the sphere is considered to 
be of infinite radius. Hence precession shows that the 
direction of the earth's axis is continually changing. 
Careful observations from the time of HippAncnrs until 
now show that the change in question consists in a slow 
revolution of the pole of the earth around the pole of the 
ecliptic as projected on the celestial sphere. The rate of 
motion is such that the revolution will be completed in 
between 25,000 and 26,000 years. At the end of this 
period the equinox and solstices will have made a com- 
plete revolution in the heavens. 

The nature of this motion will be seen more clearly by referring 
to Fig. 46, p. 109. We have there represented the earth in four 
positions during its annual revolution. We have represented the axis 
-as inclining to the right in each of these positions, and have de- 
scribed it as remaining parallel to itself during an entire revolution. 
The phenomena of precession show that this is not absolutely true, 
but that, in reality, the direction of the axis is slowly changing. 
This change is such that, after the lapse of some 6400 years, the 
north pole of the earth, as represented in the figure, will not in- 
cline to the right, but toward the observer, the amount of the in- 
clination remaining nearly the same. The result will evidently be 
a shifting of the seasons. At D we shall have the winter solstice, 
because the north pole will be inclined toward the observer and 
therefore from the sun, while at A we shall have the vernal equinox 
instead of the winter solstice, and so on. 

In 6400 years more the north pole will be inclined toward the 
left, and the seasons will be reversed. Another interval of the 
same length, and the north pole will be inclined from the observer, 
the seasons being shifted through another quadrant. Finally, at 
the end of about 25, 800 years, the axis will have resumed its original 
direction. 

Precession thus arises from a motion of the earth alone, and 
not of the heavenly bodies. Although the direction of the earth's 
axis changes, yet the position of this axis relative to the crust of the 



PRECESSION. 209 

earth remains invariable. Some have supposed that precession 
would result in a change in the position of the north pole on the 
surface of the earth, so that the northern regions would be covered 
by the ocean as a result of the different direction in which the 
ocean would be carried by the centrifugal force of the earth's rota- 
tion. This, however, is a mistake. It has been shown by a mathe- 
matical investigation that the position of the poles, and therefore 
of the equator, on the surface of the earth, cannot change except 
from some variation in the arrangement of the earth's interior. 
Scientific investigation has yet shown nothing to indicate any prob- 
ability of such a change. 

The motion of precession is not uniform, but is subject to several 
inequalities which are called Nutation. These can best be under- 
stood in connection with the forces which produce precession. 

Cause of Precession, etc. — Sir Isaac Newton showed that pre- 
cession was due to an inequality in the attraction of the sun and 
moon produced by the spheroidal figure of the earth. If the earth 
were a perfect homogeneous sphere, the direction of its axis would 




Fig. 69. 

bever change in consequence of the attraction of another body. 
But the excess of matter around the equatorial regions of the earth 
is attracted by the sun and moon in such a way as to cause a turn- 
ing force which tends to change the direction of the axis of rota- 
tion. To show the mode of action of this force, let us consider the 
earth as a sphere encircled by a large ring of matter extending 
around its equator, as in Fig. 69. Suppose a distant attracting body 
situated in the direction G c, go that the lines in which the parts of 
the ring are attracted are A a, Bb, Cc, etc., which will be nearly 
parallel. The attractive force will gradually diminish from A to 
B, owing to the greater distance of the latter from the attracting 
body. Let us put : 

r, the distance of the centre C from the attracting body, 
/o, the radius A C = B C of the equatorial ring, multiplied by the 
cosine of the angle A Cc, so that the distance of A from the attract- 
ing centre is r— p, and that of B is r + p. 
m, the mass of the attracting body ; 



210 ASTRONOMY. 

The accelerative attraction exerted at the three points A, C, B will 
then be 

m m m m 

~3 » 



(r-pY 2 r* 7 (r +p)*' 

The radius p being very small compared with r, we may develop the 

denominators of the first and third fractions in powers of — 

by the binomial theorem, and neglect all pow%rs after the first. 
The attractions will then be approximately : 

m 2«/) i m ^ m 2mp 

The forces — -— will be very small compared with — on account 

of the smallness of p. 

The principal force — will cause all parts of the body to fall 

equally toward the attracting centre, and will therefore cause no 
rotation in the body and no change in the direction of the axis 2fS. 
Supposing the body to revolve around the centre in an orbit, we 
may conceive this attraction to be counterbalanced by the so-called 
centrifugal force.* 

Subtracting this uniform principal force, there is left a force — ~ 

acting on A in the direction A a, and an equal force acting on B in 
the opposite direction ~b B. It is evident that these two forces tend 
to make the earth rotate around an axis passing through C in such 
a direction as to make the line C A m coincide with C c , and that, 
if no cause modified the action of these forces, the earth would os- 
cillate back and forth on that axis. 

* We may here mention a very common misapprehension respecting 
what is sometimes called centrifugal force, and is supposed to be a 
force tending to make a body fly away from the centre. It is some- 
times said that the body will fly from the centre when the centrifugal 
force exceeds the centripetal, and toward it in the opposite case. This is 
a mistake, such a force as this having no existence. The so-called 
centrifugal force is not properly a centrifugal force at all. but only the 
reaction of the whirling body against the centripetal force, which, by the 
third law of motion, is - equal and opposite to that force. When a stone 
is whirled in a sling the tension on the string is simply the force neces- 
sary to make the stone constantly deviate from the straight line in 
which it tends to move, and is the same as the resistance which the 
stone offers to this deviation in consequence of its inertia. So, in the 
case of the planets, the centrifugal force is only the resistance offered 
by the inertia of the planet to the sun's attraction. If the sling should 
break, or if the sun should cease to attract the planet, the centripetal 
and centrifugal forces would both cease instantly, and the stone or 
planet would, in accordance with the first law of motion, fly forward 
in the straight line in which it was moving at the moment. 



NUTATION. 211 

But a modifying cause is found in the rotation of the earth on its 
own axis, which prevents any change in the angle m C c , but 
causes a very slow revolution of the axis N 8 around the perpen- 
dicular line G B, which motion is that of precession.* 

Nutation. — It will be seen that, under the influence of the grav- 
itation of the sun and moon, precession cannot be uniform. At the 
time of the equinoxes the equator A B of the earth passes through 
the sun, and the latter lies in the line B C A m, so that the small 
precessional force tending to displace the equator must then vanish. 
This force increases on both sides of the equinox, and attains a 
maximum at the solstices when the angle m C c is 23^°. Hence the 
precession produced by the sun takes place by semi-annual steps. 
One of these steps, however, is a little longer than the other, 
because the earth is nearer the sun in December than in June. 

Again, we have seen that the inclination of the moon's orbit to 
the equator ranges from 18^* to 28|° in a period of 18-6 years. 
Since the precessional force depends on this inclination, the 
amount of precession due to the action of the moon has a period 
«qual to one revolution of the moon's node, or 18'6 years. These 
inequalities in the motion of precession are termed nutation. 

Changes in the Right Ascensions and Declinations of 
the Stars. — Since the declination of a heavenly body is its an- 
gular distance from the celestial equator, it is evident that any 
change in the position of the equator must change the declinations 
of the fixed stars. Moreover, since right ascensions are counted 
from the position of the vernal equinox 3 the change in the position 
of this equinox produced by precession and nutation must change 
the right ascensions of the stars. The motion of the equator may 
be represented by supposing it to turn slowly around an axis lying 
in its plane, and pointing to 6 h and 18 h of right ascension. All 
that section of the equator lying within 6 h of the vernal equinox 
(see Fig. 45, page 103) is moving toward the south (downward in 
the figure), while the opposite section, from 6 h to 18 h right ascen- 
sion, is moving north. The amount of this motion is 20" annually. 
It is evident that this motion will cause both equinoxes to shift 
toward the right, and the geometrical student will be able to see 
that the amount of the shift will be : 

* The reason of this seeming paradox is that the rotative forces acting 
on A and B are as it were distributed by the diurnal rotation around 
N 8. Suppose, for example, that A receives a downward and B an up- 
ward impulse, so that they begin to move in these directions. At the 
end of twelve hours A has moved around to B, so that its downward 
motion now tends to increase the angle m C c. and the upward motion of 
i?has the same effect. If we suppose a series of impulses, a diminution 
of the inclination will be produced during the first 12 hours, but after 
that the effect of each impulse will be counterbalanced by that of 12 
hours before, so that no further diminution will take place ; but 
every impulse will produce a sudden permanent change in the direction 
of the axis N 8, the end N moving toward and £ from the observer. 

This same law of rotation is exemplified in the gyroscope aud the 
child's top, each of which are kept erect by the rotation, though grav- 
ity tends to make them fall. 



212 ASTRONOMY 

On the equator, 20" cot w ; 

On the ecliptic, 20" cosec u ; 
u being the obliquity of the ecliptic (23° 2J-§'). In consequence, 
the right ascensions of stars near the equator are constantly increas- 
ing by about 46" of arc, or 3 S .07 of time annually. Away from, 
the equator the increase will vary in amount, because, owiDg to the 
motion of the pole of the earth, the point in which the equator is. 
intersected by the great circle passing through the pole and the 
star will vary as well as the equinox, it being remembered that the 
right ascension of the star is the distance of this point of intersec- 
tion from the equinox. 

The adept in spherical trigonometry will find it an improving 
exercise to work out the formulae for the annual change in the right 
ascension and declination of the stars, arising from the motion of 
the equator, and consequently of the equinox. He will find the 
result to be as follows : Put 

n, the annual angular motion of the equator (20" • 06;, 

o), its obliquity (23° 27' -5), 

a d, the right ascension and declination of the star ; 

Then we shall find : 

Annual change in R. A. = n cot cj -j- n sin a tan <J. 

Annual change in Dec. = n cos a. 



CHAPTER IX. 



CELESTIAL MEASUREMENTS OF MASS AND 
DISTANCE. 

§ 1. THE CELESTIAL SCALE OP MEASUREMENT. 

The units of length and mass employed by astronomers 
are necessarily different from those used in daily life. 
For instance, the distances and magnitudes of the heavenly 
bodies are never reckoned in miles or other terrestrial 
measures for astronomical purposes ; when so expressed 
it is only for the purpose of making the subject clearer to 
the general reader. The units of weight or mass are also, 
of necessity, astronomical and not terrestrial. The mass 
of a body may be expressed in terms of that of the sun 
or of the earth, but never in kilograms or tons, unless in 
popular language. There are two reasons for this course. 
One is that in most cases celestial distances have first to 
be determined in terms of some celestial unit — the earth's 
distance from the sun, for instance — and it is more con- 
venient to retain this unit than to adopt a new one. The 
other is that the values of celestial distances in terms of 
ordinary terrestrial units are for the most part extremelv 
uncertain, while the corresponding values in astronomical 
units are known with great accuracy. 

An extreme instance of this is afforded by the dimen- 
sions of the solar system. By a long and continued series 
of astronomical observations, investigated by means of 
Kepler's laws and the theory of gravitation, it is possible 
to determine the forms of the planetary orbits, their 
positions, and their dimensions in terms of the earth's 



214 ASTRONOMY. 

mean distance from the sun as the unit of measure, with 
great precision. It will be remembered that Kepler's 
third law enables us to determine the mean distance of a 
planet from the sun when we know its period of revolu- 
tion. Now, all the major planets, as far out as Saturn, 
have been observed through so many revolutions that their 
periodic times can be determined with great exactness — in 
fact within a fraction of a millionth part of their whole 
amount. The more recently discovered planets, Uranus 
and Neptune, will, in the course of time, have their 
periods determined with equal precision. Then, if we 
square the periods expressed in years and decimals of a 
year, and extract the cube root of this square, we have the 
mean distance of the planet with the same order of pre- 
cision. This distance is to be corrected slightly in conse- 
quence of the attractions of the planets on each other, but 
these corrections also are known with great exactness. 
Again, the eccentricities of the orbits are exactly deter- 
mined by careful observations of the positions of the plan- 
ets during successive revolutions. Thus we are enabled to 
make a map of the planetary orbits which shall be so ex- 
act that the error would entirely elude the most careful 
scrutiny, though the map itself should be many yards in 
extent. 

On the scale of this same map we could lay down the 
magnitudes of the planets with as much precision as our 
instruments can measure their angular semi-diameters. 
Thus we know that the mean diameter of the sun, as seen 
from the earth, is 32', hence we deduce from formulae 
given in connection with parallax (Chapter L, § 12), that 
the diameter of the sun is • 0093083 of the distance of the 
sun from the earth. We can therefore, on our supposed 
map of the solar system, lay down the sim in its true size, 
according to the scale of the map, from data given directly 
by observation. In the same way we can do this for each 
of the planets, the earth and moon excepted. There is 
no immediate and direct way of finding how large the 



CELESTIAL MEASURES. 215 

earth or moon would look from a planet, hence the ex- 
ception. 

But without further special research into this subject, 
we shall know nothing about the scale of our. map. It is 
clear that in order to fix the distances or the magnitudes 
of the planets according to any terrestrial standard, we 
must know this scale. Of course if we can learn either 
the distance or magnitude of any one of the planets laid 
down on the map, in miles or in semi-diameters of the 
earth, we shall be able at once to find the scale. But this 
process is so difficult that the general custom of astrono- 
mers is not to attempt to use an exact scale, but to employ 
the mean distance of the sun from the earth as the unit in 
celestial measurements. Thus, in astronomical language, 
we say that the distance of Mercury from the sun is 
0-387, that of Venus 0-723, that of Mars 1-523, that 
of Saturn 9 • 539, and so on. But this gives us no in- 
formation respecting the distances and magnitudes in terms 
of terrestrial measures. The unknown quantities of our 
map are the magnitude of the earth on the scale of the 
map, and its distance from the sun in terrestrial units of 
length. Could we only take up a point of observation 
from the sun or a planet, and determine exactly the angu- 
lar magnitude of the earth as seen from that point, we 
should be able to lay down the earth of our map in its cor- 
rect size. Then since we already know the size of the 
earth in terrestrial units, we should be able to find the 
scale of our map, and thence the dimensions of the whole 
system in terms of those units. 

It will be seen that what the astronomer really wants is 
not so much the dimensions of the solar system in miles as 
to express the size of the earth in celestial measures. 
These, however, amount to the same thing, because hav- 
ing one, the other can be readily deduced from the known 
magnitude of the earth in terrestrial measures. 

The magnitude of the earth is not the only unknown 
quantity on our map. From Kepler's laws we can de- 



216 ASTRONOMY. 

termine nothing respecting the distance of the moon from 
the earth, because unless a change is made in the units of 
time and space, they apply only to bodies moving around 
the sun. "We must therefore determine the distance of 
the moon as well as that of the sun to be able to complete 
our map on a known scale of measurement. 

§ 2. MEASURES OF THE SOLAR PARALLAX. 

The problem of distances in the solar system is reduced 
by the preceding considerations to measuring the distances 
of the sun and moon in terms of the earth's radius. The 
most direct method of doing this is by determining their 
respective parallaxes, which we have shown to be the same 
as the earth's angular semi-diameter as seen from them, 
in the case of the sun, the required parallax can be de- 
termined as readily by measuring the parallaxes of any 
of the planets as by measuring that of the sun, because 
any one measured distance on the man will give us the 
scale of our map. Now, the planets Venus and liars oc- 
casionally come much nearer the earth than the sun ever 
does, and their jDarallaxes also admit of more exact meas- 
urement. The parallax of the sun is therefore determined 
not by observations on the sun itself, but on these two 
planets. Three methods of finding the sun's parallax in 
this way have been applied. They are : 

(1.) Observations of Venus in transit across the sun. 

(2.) Observations of the declination of Mars from 
widely separated stations on the earth's surface. 

(3.) Observations of the right ascension of Mars, near 
the times of its rising and setting, at a single station. 

Solar Parallax from Transits of Venus. — The general 
principles of the method of determining the parallax of a 
planet by simultaneous observations at distant stations 
will be seen by referring to Fig. 18, p. 49. If two ob- 
servers, situated at S f and S\ make a simultaneous ob- 
servation of the direction of the body P s it is evident 



TRANSITS OF VENUS. 217 

that the solution of a plane triangle will give the distance 
of P from each station. In practice, however, it would 
be impracticable to make simultaneous observations at 
distant stations, and as the planet is continually in motion, 
the problem is a much more complex one than that of 
simply solving a triangle. The actual solution is effected 
by a process which is algebraic rather than geometrical, 
but we may briefly describe the geometrical nature of the 
problem. 

Considering the problem as a geometrical one, it is evi- 
dent that, owing to the parallax of Venus being nearly four 
times as great as that of the sun, its path across the sun's 
disk will be different when viewed from different points of 
the earth's surface. The further south we go, the further 
north the planet will seem to be on the sun's disk. The 
change will be determined by the difference between the 
parallax of Venus and that of the sun, and this makes the 
geometrical explanation less simple than in the case of a 
determination into which only one parallax enters. It 
will be sufficient if the reader sees that when we know the 
relation between the two parallaxes — when, for instance, 
we know that the parallax of Venus is 3-78 times that of 
the sun — the observed displacement of Venus on the sun's 
disk will give us both parallaxes. The u relative paral- 
lax," as it is called, will be 2-78 times the sun's parallax, 
and it is on this alone that the displacement depends. . 

The algebraic process, which is that actually employed in the 
solution of astronomical problems of this class, is as follows : 

Each observer is supposed to know his longitude and lati- 
tude, and to have made one or more observations of the angular 
distance of the centre of the planet from the centre of the sun. 

To woik up the observations, the investigator must have an 
ephemerls of Venus and of the sun — that is, a table giving 
the right ascension and declination of each body from hour to hour 
as calculated from the best astronomical data. The ephemeris can 
never be considered absolutely correct, but its error may be as- 
sumed as constant for an entire day or more. By means of it, the 
right ascension and declination of the planet and of the sun, as seen 
from the centre of the earth, may be computed at any time. 

It is shown in works on spherical astronomy how, when the right 



218 ASTRONOMY. 

ascensions, declinations, and parallaxes of Venus and the sun are 
given for a definite moment, the distance of their centres, as seen 
from a given point on the surface of the earth, may be computed. 
Referring to such works for the complete demonstration of the re- 
quired formulas, we shall give the approximate results in such a way 
as to show the principle involved. Let us put : 

a, 6, p, the geocentric right ascension and declination of Venus, as 
given in the ephemeris for the moment of observation, and its dis- 
tance from the earth's centre. 

a, 6' , p , the same quantities for the sun. 

7r, the sun's equatorial horizontal parallax at distance unity. 

H, the hour angle of the sun, as seen from the place at the mo- 
ment of observation. 

f', the geocentric latitude of the observer. 

r, the earth's radius at the point of observation ; that is, the dis- 
tance of the observer from the earth's centre, the equatorial radius 
being taken as unity. 

The parallax is so small that we may regard it as equal to its sine. 
If we put : 

7Ti, the equatorial horizontal parallax of Venus at its actual dis- 
tance, p ; 

rc-'i, the same for the sun. 

Then, because the parallaxes are inversely as the distances of the 
bodies : 

p P 

If we put : 

D, the angular distance of the centres of Venus and the sun, as 
seen from the earth's centre, D will be the hypothenuse of a nearly 
right-angled spherical triangle, of which the north and south side 
will be the difference of declination ; and the east and west side the 
difference of right ascension, multiplied by the cosine of the decli- 
nation. We shall, therefore, have approximately: 

D* = (6 — d') 2 + (a — af cos. 2 6'. (2) 

This value of B being very near the truth, it is supposed that the 
effect of small corrections to a, a' , 6 and 6' may be treated as differ- 
entials, and obtained by differentiation. Differentiating the above 
expression, and dividing by 2, we have : 

DdB = {6—6') (d6 — d6') + (a — a') cos. 2 <5' {da — da) 

dD — — — (d6 - d6')-L- a ~° cos. 2 6' (da — da). (31 

D D 

Because the observer is at the earth's surface the apparent direc- 
tion of the two bodies, and hence the values of a, a', 6 and 6', will 
be changed by parallax. If we suppose the differentials, da, dd, etc., 
to represent the changes due to parallax, it is shown in spherical as- 
tronomy that they may be computed by the formulae : 



TRANSITS OF VENUS. 



219 



da = r cos. < 
da = r cos. < 
dd = (r cos. 
dd' = (r cos. 



sec. 6 sin. H X n't, 
sec. J' sin. H' X t'i, 
sin. d cos. H — r sin. 
' sin. J' cos. H — r sii 



cos. <J) X *n, 
0' cos. <T) X 7t'i« 



(4) 



The quantities in the second members of these equations are all 
supposed to be known, at least within their thousandth part, ex- 
cept 77, and 71-', the parallaxes. Moreover since the distances p and 
p are also known, if we substitu'e the values of tti, and tt'i, from the 
equations (1), 7r, the mean parallax of the sun itself, will be the only 
unknown quantity left. So if we put, for brevity, 

r cos. (j)' sec. 6 sin. H 

a = 



b = 







P 










r cos. 


<?' 


sec 


d' sin 


# 










P 




5 




r cos. 


f 


sin. 


<5 cos. 


tf- 


r sin. ^' 


cos. d 










P 






r cos. 


<P' 


sin. 


d' cos. 


II - 


r sin. ^' 


cos. (5' 



(5) 



(6) 



b' = 

P 

we have, for the effect of parallax, 

da ■=. ccr ; da' — a\ 
dd = bir ; <W = 6'7r. 

If there were no parallax, and if the values of the right ascension 
and declination given in the ephemeris were perfectly correct, the 
values of D computed from (2) would be those given' by a coi rect 
measurement from any point of the earth's surface. Suppose that 
the observer on measuring the value of D, finds it different from 
that calculated. Assuming his measure to be correct, he must as- 
sume the difference to be due to two causes : 

Firstly, parallax ; 

Secondly, errors in the values of a and 6 given in the ephemeris. 

For the effect of parallax we substitute in (3) the values of da, 
etc., in (6). We thus have : 



dB- 



D 



(b-b') + 



I) 



cos. 2 d' (a — a') 



(7) 



In this equation all the quantities in the second niemb r ex- 
cept rr are supposed to be known, and we may represent the co- 
efficient of 7T by the single symbol c, putting: 

dD — ct (8) 

To consider the effect of the second cause we must suppose dd, 
dd', da and da' in (3) to be replaced by dd dd', da and da\ which we 
put for the unknown corrections to the positions in the ephemeris. 
If we put, for brevity, 



220 ASTRONOMY. 

y = 88 — 88' ; a? = 6a — <5a' 

(5 — 8' a — a' 



D D 



cos. 2 8' ■=. n 



we shall have 



dD = my + nx. (9) 

The true value of D is given by adding the two values of dD in (8) 
and (9) to the value of D computed from (2). Hence, this true 
value of D is 

D + my + nx + cir, (10) 

in which D, m, n and c are all calculated numbers, and x, y and n- are 
unknown. 

Now, suppose that, at this same moment, the observer has meas- 
sured the distance of the centres of the two bodies and found it to 
be D . This being supposed true, must be equal to (10), that is, we 
must have 

D -+- my + nx + ctt =z D' ; 

or, by transposing, 

my + nx + ctt = D' — D. 

Thus, for every observation of distance, we have an equation of 
condition between the three unknown quantities y, x and tc. The 
solution of these equations gives the value of x, y and w, the un- 
known quantities required. 

Measurements of the Parallax of Mars.— This parallax may 
be determined from observations in two ways. In that usually 
adopted there are two observers or sets of observers, one in the 
northern and the other in the southern hemisphere, each of whom 
determines the declination of the planet from day to day at the 
moment of transit over his meridian. These declinations will be 
different by the whole amount of parallactic difference between the 
two stations, or by the angle S' PS" in Fig. 18, p. 49. The observa- 
tions are continued through the period when Mars is nearest the earth, 
generally about a couple of months. Any opposition of the planet 
may be chosen for this purpose, but the most favorable ones are 
those when the planet is nearest its perihelion. Should the planet 
be exactly at its perihelion at the time of opposition, its distance 
from the earth would be only about 0-37, while at aphelion it would 
be 0-68. This great difference is owing to the considerable eccen- 
tricity of the orbit of Mars, as can be seen by studying Fig. 48, 
p. 115, which gives a plan of most of the orbits of the larger planets. 
The favorable oppositions occur at intervals of 15 or 17 years. One 
was that of 1862, which gave almost the first conclusive evidence that 
the old parallax of the sun found by Encke was too small. This 
parallax was 8""577, and the corresponding distance of the sun was 
95£ millions of miles. The observations of 1862 seemed to show 
that this parallax must be increased by about one thirtieth part, and 
the distance diminished in about the same ratio. But the most recent 
results make it probable that the change should not be quite so 
great as this. 



PARALLAX OF MARS. 221 

Parallax of Mars in Right Ascension.— Another method 
of measuring the parallax of Mars is founded on principles entirely 
different from those we have hitherto conside: ed . In the latter, 
observations have to be made by two observers in opposite hemi- 
spheres of the earth. But an observer at any point on the earth's 
surface is carried around on a circle of latitude every day by the 
diurnal motion of the earth. In consequence of this motion, there 
must be a corresponding apparent motion of each of the planets in 
an opposite direction. In other words, the parallax of the planet 
must be different at different times of the day. This diurnal 
change in the direction of the planet admits of being measured in 
the following way : The effect of parallax is always to make a 
heavenly body appear nearer the horizon than it would appear as seen 
from the centre of the earth. This will be obvious if we reflect 
that an observer moving rapidly from the centre of the earth to its 
circumference, and keeping his eye fixed upon a planet, would see 
the planet appear to move in an opposite direction — that is, down- 
ward relative to the point of the earth's surface which he aimed at. 
Hence a planet rising in the east will rise later in consequence of 
parallax, and will set earlier. Of course the rising and setting 
cannot be observed with sufficient accuracy for the purpose of 
parallax, but, since a fixed star has no parallax, the position of 
the planet relative to the stars in its neighborhood will change 
during the interval between the rising and setting of the planet. 
The observer therefore determines the positon of Mars relative 
to the stars surrounding him shortly after he rises and again 
shortly before he sets. The observations are repeated night 
after night as often as possible. Between each pair of east and 
west observations the planet will of course change its position 
among the stars in consequence of the orbital motions of the 
earth and planet, but these motions can be calculated and allowed 
for, and the changes still outstanding will then be due to parallax. 

The most favorable regions for an observer to determine the par- 
allax in this way are those near the earth's equator, because he is 
there carried around on the largest circle. If he is nearer the poles 
than the equator, the circle will be so small that the parallax will be 
hardly worth determining, while at the poles there will be no par- 
allactic change at all of the kind just described. 

Applications of this method have not been very numerous, 
although it was suggested by Flamsteed nearly two centuries ago. 
The latest and most successful trial of it was made by Mr. David Gill 
of England during the opposition of Mars in 1877 above described. 
The point of observation chosen by him was the island of Ascen- 
sion, west of Africa and near the equator. His measures indicate 
a considerable reduction in the recently received values of the solar 
parallax, and an increase in the distance of the sun, making the 
latter come somewhat nearer to the old value. 

Accuracy of the Determinations of Solar Parallax. 

The parallax of Mars at opposition is rarely more than 



222 ASTRONOMY, 

20", and the relative parallax of Venus and the sun at the 
time of the transit is less than 24". These quantities are 
so small as to almost elude very precise measurement ; it 
is hardly possible by any one set of measures of parallax 
to determine the latter without an uncertainty of -g-J-g- of its 
whole amount. In the distance of the sun this corre- 
sponds to an uncertainty of nearly half a million of miles. 
Astronomers have therefore sought for other methods of 
determining the sun's distance. Although some of these 
may be a little more certain than measures of parallax, there 
is none by which the distance of the sun can be determined 
with any approximation to the accuracy which character- 
izes other celestial measures. 

Other Methods of Determining Solar Parallax. — A 
very interesting and probably the most accurate method 
of measuring the sun's distance is by using light as a mes- 
senger between the sun and the earth. We shall hereafter 
see, in the chapter on aberration, that the time required for 
light to pass from the sun to the earth is known with con- 
siderable exactness, being very nearly 498 seconds. If 
then we can determine experimentally how many miles or 
kilometres light moves in a second, we shall at once have 
the distance of the sun by multiplying that quantity by 
498. But the velocity of light is about 300,000 kilometres 
per second. This distance would reach about eight times 
around the earth. It is rarely possible that two points on 
the earth's surface more than a hundred kilometres apart 
are visible from eacli other, and distinct vision at distances 
of more than twenty kilometres is rare. Hence to deter- 
mine experimentally the time required for light to pass 
between two terrestrial stations requires the measurement of 
an interval of time, which even under the most favorable 
cases can be only a fraction of a thousandth of a second. 
Methods of doing it, however, have been devised and ex- 
ecuted by the French physicists, Fizeau, Foucault, and 
Coenu, and quite recently by Ensign Michelson at the 
U. S. Naval Academy, Annapolis. From the experiments 



80LAB PARALLAX. 223 

of the latter, which are probably the most accurate, the 
velocity of light would seem to be about 299,900 kilome- 
tres per second. Multiplying this by 498, we obtain 149,- 
350,000 kilometres for the distance of the sun. The time 
required for light to pass from the sun to the earth is still 
uncertain by nearly a second, but this value of the sun's 
distance is probably the best yet obtained. The corre- 
sponding value of the sun's parallax is 8" -81. 

Yet other methods of determining the sun's distance 
are given by the theory of gravitation. The best known 
of these depends upon the determination of the parallactic 
inequality of the moon. It is found by mathematical in- 
vestigation that the motion of the moon is subjected to 
several inequalities, having the sun's horizontal parallax 
as a factor. In consequence of the largest of these in- 
equalities, the moon is about two minutes behind its mean 
place near the first quarter, and as far in advance at the 
last quarter. If the position of the moon could be deter- 
mined by observation with the same exactness that the po- 
sition of a star or planet can, this would probably afford 
the most accurate method of determining the solar par- 
allax. But an observation of the moon has to be made, 
not upon its centre, but upon its limb or circumference. 
Only the limb nearest the sun is visible, the other one 
being unilluminated, and thus the illuminated limb on 
which the observation is to be made is different at the first 
and third quarter. These conditions induce an uncertain- 
ty in the comparison of observations made at the two 
quarters which cannot be entirely overcome, and therefore 
leave a doubt respecting the correctness of the result. 

Brief History of Determinations of the Solar Parallax. 
— The distance of the sun must at all times have been one 
of the most interesting scientific problems presented to the 
human mind. The first known attempt to effect a solu- 
tion of the problem was made by Akistakchus, who nour- 
ished in the third century before Christ. It was founded 
on the principle that the time of the moon's first quarter 



224 ASTRONOMY. 

will vary with the ratio between the distance of the moon 
and sun, which may be shown as follows. In Fig. 70 
let jE* represent the earth, M the moon, and S the sun. 
Since the sun always illuminates one half of the lunar 
globe, it is evident that when one half of the moon's disk 
appears illuminated, the triangle E M S must be right- 
angled at M. The angle M E S can he determined by 
measurement, being equal to the angular distance between 
the sun and the moon. Having two of the angles, the 
third can be determined, because the sum of the three 
must make two right angles. Thence we shall have the 
ratio between EM, the distance of the moon, and E S> 
the distance of the sun, by a trigonometrical computation. 




Fig. 70. 

Then knowing the distance of the moon, which can be 
determined with comparative ease, we have the distance of 
the sun by multiplying by this ratio. Aristarciius con- 
cluded, from his supposed measures, that the angle M ES 
w T as three degrees less than a right angle. We should 

then have -=-^- = sin 3° = T 1 ¥ very nearly. It would 

follow from this that the sun was 19 times the distance 
of the moon. We now know that this result is entirely 
wrong, and that it is impossible to determine the time 
when the moon is exactly half illuminated with any ap- 
proach to the accuracy necessary in the solution of the 
problem. In fact, the greatest angular distance of the 



SOLAR PARALLAX. 225 

earth and moon, as seen from the sun — that is, the angle 
ES M — is only about one quarter the angular diameter of 
the moon as seen from the earth. 

The second attempt to determine the distance of the 
sun is mentioned by Ptolemy, though Hipparchus may be 
the real inventor of it. It is founded on a somewhat com- 
plex geometrical construction of a total eclipse of the 
moon. It is only necessary to state the result, which 
was, that the sun was situated at the distance of 1210 radii 
of the earth. This result, like the former, was due only 
to errors of observation. So far as all the methods known 
at the time could show, the real distance of the sun ap- 
peared to be infinite, nevertheless Ptolemy's result was 
received without question for fourteen centuries. 

When the telescope was invented, and more accurate 
observations became possible, it was found that the sun's 
distance must be greater and its parallax smaller than 
Ptolemy had supposed, but it was still impossible to give 
any measure of the parallax. All that could be said was 
that it was less than the smallest quantity that could be de- 
cided on by measurement. The first approximation to the 
true value was made by Horrox of England, and after- 
ward by Huyghens of Holland. It was not founded on 
any attempt to measure the parallax directly, but on an 
estimate of the probable magnitude of the earth on the 
scale of the solar system. The magnitude of the planets 
on this scale being known by measurement of their appar- 
ent angular diameters as seen from the earth, the solar 
parallax may be found when we know the ratio between 
the diameter of the earth and that of any planet whose 
angular diameter has been measured. Now, it was sup- 
posed by the two astronomers we have mentioned that 
the earth was probably of the same order of magnitude 
with the other planets. 

Horrox had a theory, which we now know to be erro- 
neous, that the diameters of the planets were proportional 
to their distances from the sun — in other words, that all 



226 ASTRONOMY. 

the planets would appear of the same diameter when seen 
from the sun. This diameter he estimated at 28", from 
which it followed that the solar parallax was 14:". Huyghens 
assumed that the actual magnitude of the earth was mid- 
way between those of the two planets Venus and Mars on 
each side of it ; he thus obtained a result remarkably near 
the truth. It is true that in reality the earth is a little 
larger than either Venus or Mars, but the imperfect tel- 
escopes of that time showed the planets larger than they 
really were, so that the mean diameter of the enlarged 
planets, as seen in the telescope of Huyghens, was such as 
to correspond very nearly to the diameter of the earth. 

The first really successful measure of the parallax 
of a planet was made upon Mars during the opposition of 
1672, by the first of the two methods already described. 
An expedition was sent to the colony of Cayenne to ob- 
serve the declination of the planet from night to night, 
while corresponding observations were made at the Paris 
Observatory. From a discussion of these observations, 
Cassini obtained a solar parallax of 9" -5, which is within 
a second of the truth. The next steps forward were made 
by the transits of Venus in 1761 and 1769. The leading 
civilized nations caused observations on these transits to be 
made at various points on the globe. The method used 
was very simple, consisting in the determination of the 
times at which Venus entered upon the sun's disk and left 
it again. The absolute times of ingress and egress, as seen 
from different points of the globe, might differ by 20 
minutes or more on account of parallax. The results, 
however, were found to be discordant. It w r as not until 
more than half a century had elapsed that the observations 
were all carefully calculated by Encke of Germany, wlio 
concluded that the parallax of the sun was 8" • 578, and the 
distance 95 millions of miles. 

In 1854 it began to be suspected that Encke' s value of 
the parallax was much too small, and great labor was now 
devoted to a solution of the problem. Hansen, from the 



MASSES OF THE SUN AND EARTH. 227 

parallactic inequality of the moon, first found the parallax 
of the sun to be 8" -97, a quantity which he afterward re- 
duced to 8" -916. This result seemed to be confirmed by 
other observations, especially those of Mars during the 
opposition of 1862. It was therefore concluded that the 
sun's parallax was probably between 8" -90 and 9" -00. 
Subsequent researches have, however, been diminishing 
this value. In 1867, from a discussion on all the data 
which were considered of value, it was concluded by one 
of the writers that the most probable parallax was 8" • 848. 
The measures of the velocity of light made by Michelson 
in 1878 reduce this value to 8" -81, and it is now doubtful 
whether the true value is any larger than this. 

The observations of the transit of Venus in 1874 have 
not been completely discussed at the time of writing these 
pages. "When this is done some further light may be 
thrown upon the question. It is, however, to the deter- 
mination of the velocity of light that we are to look for 
the best result. All we can say at present is that the so- 
lar parallax is probably between 8" -79 and 8" -83, or, if 
outside these limits, that it can be very little outside. 



§ 3. RELATIVE MASSES OF THE SUN AND 
PLANETS. 

In estimating celestial masses as well as distances, it is necessary 
to use what we may call celestial units — that is, to take the mass of 
some celestial body as a unit, instead of any multiple of the pound 
or kilogram. The reason of this is that the ratios among the 
masses of the planetary system, or, which is the same thing, the 
mass of each body in terms of that of some one body as the unit, 
can be determined independently of the mass of any one of th<_jn. 
To express a mass in kilograms or other terrestrial units, it is neces- 
sary to find the mass of the earth in such units, as already explained. 
This, however, is not necessary for astronomical purposes, where only 
the relative masses of the several planets are required. In estimat- 
ing the masses of the individual planets, that of the sun is generally 
taken as a unit. The planetary masses will then all be very small 
fractions. 

Masses of the Earth and Sun.— We shall first consider the 
mass of the earth because it is connected by a very curious relation 
with the parallax of the sun. Knowing the latter, we can determine 



228 ASTRONOMY. 

the mass of the sun relative to the earth, which is the same thing 
as determining the astronomical mass of the earth, that of the sun 
being unity. This may be clearly seen by reflecting that when we 
know the radius of the earth's orbit we can determine how far the 
earth moves aside from a straight line in one second in consequence 
of the attraction of the sun. This motion measures the attractive 
force of the sun at the distance of the earth. Comparing it with 
the attractive force of the earth, and making allowance for the 
difference of distances from centres of the two bodies, we deter- 
mine the ratio between their masses. 

The calculation in question is made in the most simple and ele- 
mentary manner as follows. Let us put : 

rr, the ratio of the circumference of a circle to its diameter (tt = 
3-14159 ...) 

r, the mean radius of the earth, or the radius of a sphere having 
the same volume as the earth. 

a, the mean distance of the earth from the sun. 

g, the force of gravity on the earth's surface at a point where the 
radius is r — that is, the distance which a body will fall in one 
second. 

g f , the sun's attractive force at the distance a. 

T, the number of seconds in a sidereal year. 

M, the mass of the sun. 

m, the mass of the earth. 

P, the sun's mean horizontal parallax. 

The force of gravity of the sun, g\ may be considered as equal to 
the so-called centrifugal force of the earth, or to the distance which 
the earth falls toward the sun in one second. By the formula for 
centrifugal force given in Chapter VIII., p. 204, we have, 

, _ 4 tt 2 a 
and by the law of gravitation, 



whence 




and 

M 



4 TT 2 O/ 



yr2 

We have, in the same way, for the earth, 



m 
3> 



9 = 

whence 

m = gr 1 



MASS OF THE SUN. 229 

Therefore, for the ratio of the masses of the earth and sun, we have : 

M 4 7T 2 a 3 4 7T 2 r a 3 , N 

m gT* r 2 T" g r 3 w 

By the formulae for parallax in Chapter I., §8, we have: 

.a 3 1 

r = asm P . \ — 



,- 3 sin 3 P 
Therefore 

m T 2 ' g ' sin 3 P { h 

The quantities T, rand g may be regarded as all known with great 
exactness. We see that the mass of the earth, that of the sun being 
unity, is proportional to the cube of the solar parallax. 

From data already given, we have : 

T= 365 days, 6 hours, 9 nl 9 s ; in seconds, T— 31 538 149, 
Mean radius of the earth in metres,* . . r = 6 370 008, 
Force of gravity in metres, . . . . g= 9-8202, 

while log 7T 2 = 1-59636. Substituting these numbers in the formulae, 
it may be put in the form, 

Z^ = [7-58984] sn»P,t 

where the quantity in brackets is the logarithm of the factor. 

It will be convenient to make two changes in the parallax P. This 
angle is so exceedingly small that we may regard it as equal to its 
sine. To express it in seconds we must multiply it by the number 
of seconds in the unit radius — that is, by 206265 ". This will make 
P (in seconds) = 206265 "sin P. Again, the standard to which par- 
allaxes are referred is always the earth's equatorial radius, which is 
greater than r by about ^ of its whole amount. So, if we put P" 
for the equatorial horizontal parallax, expressed in seconds, we shall 
have, 

P' == (1 + ^) 206265" sin P = [5 - 31492] shi P, 
whence, for sin P in terms of P'\ 

sinP = — ~ 

[5-31492] 

* The mean radius of the earth is not the mean of the polar and 
equatorial radii, but one third the sum of the polar radius and twice 
the equatorial one, because we can draw three such radii, each mak- 
ing a right angle with the other two. 

f A number enclosed in brackets is frequently used to signify the 
logarithm af a coefficient or divisor to be used. 



?! 



230 



ASTRONOMY. 



If we substitute this value in the expression for the quotient of 
the masses, it may be put into either of the forms : 



M _ [8- 35493 J 



P"=[2-78498j f~^\ 



MJ 

The first formula gives the ratio of the masses when the solar par- 
allax is known ; the second, the parallax when the ratio of the masses 
is known. The following table shows, for different values of the 
solar parallax, the corresponding ratio of the masses, and distance of 
the sun in terrestrial measures : 





M 

m 


Distance op the Sun. 


Solar 

Parallax. 

P" 


In equatorial 

radii of the 

earth. 


In millions of 
miles. 


In millions of 
kilometres. 


8"- 75 
8". 76 
8". 77 
8". 78 
8"- 79 
8". 80 
8"- 81 
8" -82 
8". 83 
8" -84 
8" -85 


337992 
336835 
335684 
334538 
333398 
332262 
331132 
330007 
328887 
327773 
326664 


23573 
23546 
23519 
23492 
23466 
23439 
23413 
23386 
23360 
23333 
23307 


93 
93 
93 
93 
92 
92 
92 
92 
92 
92 
92 


421 
314 
208 
102 
996 
890 
785 
680 
575 
470 
366 


150-343 

150-172 

150-001 

149-830 

149-660 

149-490 

149-320 

149-151 

148-982 • 

148-814 

148-646 



We have said that the solar parallax is probably contained between 
the limits 8". 79 and 8". 83. It is certainly hardly more than one or 
two hundredths of a second without them. So, if we wish to express 
the constants relating to the sun in round numbers, we may say that — 

Its mass is 330,000 times that of the earth. 

Its distance iu miles is 93 millions, or perhaps a little less. 

Its distance in kilometres is probably between 149 and 150 mil- 
lions. 

Density of the Sun. — A remarkable result of the preceding 
investigation is that the density of the sun, relative to that of the 
earth, can be determined independently of the mass or distance of 
the sun by measuring its apparent angular diameter, and the force 
of gravity at the earth's surface. Let us put 

7), the density of the sun. 

d, that of the earth. 

s, the sun's angular semi-diameter, as seen from the earth. Then, 
continuing the notation already given, we shall have : 



MASS OF THE SUN. 
Linear radius of the sun = a sin s. 
Volume of the sun 



231 



— a 3 sin 3 s 
3 



(from the formula for the volume of a sphere). 



4tt 



Mass of the sun, M = -— a % D sin 8 s. 



4:77 



Mass of the earth, m = -^-r a > 



Substituting these values of M and m in the equation (a), and 
dividing out the common factors, it will become 



D 



47rV 



js ,n *=y?p 

from which we find, for the ratio of the density of the earth to that 
of the sun, 

d 9 T * • 3 

This equation solves the problem. But the solution may be trans- 
formed in expression. We know from the law of falling bodies that 
a heavy body will, in the time t, fall through the distance \ g t 2 . 
Hence the factor g T 2 is double the distance which a body would fall 
in a sidereal year, if the force of gravity could act upon it continu- 
ously with the same intensity as at the surface of the earth. Hence 

—z — will be the number of radii of the earth through which the 



body will fall in a sidereal year, 
preceding equation will become, 



If we put F for this number, the 



i^sin 5 



2tt' 



We therefore have this rule for finding the density of the earth 
relative to that of the sun : 

Find how many radii of the earth a heavy body would fall through 
in a sidereal year in virtue of the force of gravity at the earth's sur- 
face. Multiply this number by the cube of the sine of the sun's angular 
semi-diameter, as seen from the earth, and divide by the numerical 
factor 2 n 2 - =- 19-7392. The quotient will be the ratio of the density 
of the earth to that of the sun. 

From the numerical data already given, we find : 

Density of earth, that of sun being unity, 



D 



= 3-9208. 



232 ASTRONOMY. 

Density of the sun, that of the earth being unity, 

:? = 0-25505. 
a 

These relations do not give us the actual density of either body. 
"We have said that the mean density of the earth is about 5f, that of 
water being unity. The sun is therefore about 40 or 50 per cent 
denser than water. 

Masses of the Planets.— If we knew how far a body would 
fall in one second at the surface of any other planet than the earth, 
we could determine its mass in much the same way as we have de- 
termined that of the earth. Now if the planet has a satellite re- 
volving around it, we can make this determination — not indeed 
directly on the surface of the planet, but at the distance of the sat- 
ellite, which will equally give us the required datum. Indeed by 
observing the periodic time of a satellite, and the angle subtended by 
the major axis of its orbit around the planet, we have a more direct 
datum for determining the mass of the planet than we actually have 
for determining that of the earth. (Of course we here refer to the 
masses of the planets relative to that of the sun as unity.) In fact 
could an astronomer only station himself on the planet Venus and 
make a series of observations of the angular distance of the moon 
from the earth, he could determine the mass of the earth, and 
thence the solar parallax, with far greater precision than we are like- 
ly to know it for centuries to come. Let us again consider the 
equation for M found on page 228 : 



Here a and T may mean the mean distance and periodic time of 

a? 
any planet, the quotient — - being a constant by Kepler's third 

law. In the same equation we may suppose a the mean distance of 
a satellite from its primary, and T its time of revolution, and M wilL 
then represent the mass of the planet. "We shall have therefore for 
the mass of the planet, 

4tt 2 « /3 

a' being the mean distance of the satellite from the planet, and T* 
its time of revolution. Therefore, for the mass of the planet rel 
ative to that of the sun we have : 



M ~ a 3 T' 2 ' 

Let us suppose a to be the mean distance of the planet from the 
sun, in which case T must represent its time of revolution. Then, 
if we put s for the angle subtended by the radius of the orbit of the 



MASSES OF THE PLANETS. 233 

satellite, as seen from the sun, we shall have, assuming the orbit 
to be seen edgewise, 

a' 

sin s = — 
a 

If the orbit is seen in a direction perpendicular to its plane, we 
should have to put tang s for sin s in this formula, but the angle 
s is so small that the sine and tangent are almost the same. If we 
put t for the ratio of the time of revolution of the planet to that of 
the satellite, it will be equivalent to supposing 

_T_ 

T ~Y r 

The equation for the mass of the planet will then become 

m 2-3 

M~ SmS ' 

which is the simplest form of the usual formula for deducing the 
mass of a planet from the motion of its satellite. It is true that we 
cannot observe s directly, since we cannot place ourselves on the 
.sun, but if we observe the angle s from the earth we can always 
reduce it to the sun, because we know the proportion between the 
distances of the planet from the earth and from the sun. 

All the large planets outside the earth have satellites ; we can 
therefore determine their masses in this simple way. The earth 
having also a satellite, its mass could be determined in the same 
way but for the circumstance already mentioned that we cannot 
determine the distance of the moon in planetary units, as we can 
the distance of the satellites of the other planets from their pri- 
maries. 

The planets Mercury and Venus have no satellites. It is therefore 
necessary to determine their masses by their influence in altering 
the elliptic motions of the other planets round the sun. The altera- 
tions thus produced are for the most part so small that their deter- 
mination is a practical problem of some difficulty. Thus the action 
of Mercury on the neighboring planet Venus rarely changes the po- 
sition of the latter by more than one or two seconds of arc, unless 
we compare observations more than a century apart. But regular 
and accurate observations of Venus were rarely made until after the 
beginning of this century. The mass of Venus is best determined 
by the influence of the planet in changing the position of the plane 
of the earth's orbit. Altogether, the determination of the masses 
of Mercury and Venus presents one of the most complicated prob- 
lems with which the mathematical astronomer has to deal. 



CHAPTER X. 

THE REFRACTION AND ABERRATION OF LIGHT. 

§ 1. ATMOSPHERIC REFRACTION. 

When we refer to the place of a planet or star, we 
usually mean its true place — i.e., its direction from 
an observer situated at the centre of the earth, consid- 
ered as a geometrical point. We have shown in the sec- 
tion on parallax how observations which are necessarily 
taken at the surface of the earth are reduced to what they 
would have been if the observer were situated at the 
earth's centre. In this, however, we have supposed the 
star to appear to be projected on the celestial sphere in 
the prolongation of the line joining the observer and the 
star. The ray from the star is considered as if it suffered 
no deflection in passing through the stellar spaces and 
through the earth's atmosphere. But from the principles 
of physics, we know that such a luminous ray passing from 
an empty space (as the stellar spaces are), and through an 
atmosphere, must suffer a refraction, as every ray of light 
is known to do in passing from a rare into a denser 
medium. As we see the star in the direction which its 
light beam has when it enters the eye — that is, as we pro- 
ject the star on the celestial sphere by prolonging this 
light beam backward into space — there must be an appar- 
ent displacement of the star from refraction, and it is 
this which we are to consider. 

We may recall a few definitions from physics. The 
ray which leaves the star and impinges on the outer sur- 



REFRACTION. 235 

face of the earth's atmosphere is called the incident ray ; 
after its deflection by the atmosphere it is called the re- 
fracted ray. The difference between these directions is 
called the astronomical refraction. If a normal is drawn 
(perpendicular) to the surface of the refracting medium at 
the point where the incident ray meets it, the acute angle 
between the incident ray and the normal is called the 
angle of incidence, and the acute angle between the nor- 
mal and the refracted ray is called the angle of refraction. 
The refraction itself is the difference of these angles. 
The normal and both incident and refracted rays are in 
the same vertical plane. In 
Fig. 71 S A is the ray incident 
upon the surface B A of the re- 
fracting medium B' B A JV, 
A C is the refracted ray, M N 
the normal, SAM and CAN 
the angles of incidence and re- 
fraction respectively. Produce 
C A backward in the direction 
AS' : S A S is the refraction. 
An observer at C will see the 

r, ./• • ™ a ry, FIG ' 71 . —REFRACTION. 

star 6 as it it were at S , A S 

is the apparent direction of the ray from the star S, and 
S' is the apparent place of the star as affected by refrac- 
tion. 

This supposes the space above B B' in the figure to be 
entirely empty space, and the earth's atmosphere, equally 
dense throughout, to fill the space below B B '. In fact, how- 
ever, the earth's atmosphere is most dense at the surface of 
the earth, and gradually diminishes in density to its exterior 
boundary. Therefore, if we wish to represent the facts as 
they are, we must suppose the atmosphere to be divided 
into a great number of parallel layers of air, and by as- 
suming an infinite number of these we may also assume that 
throughout each of them the air is equally dense. Hence 
the preceding figure will only represent the refraction, at 




236 ASTRONOMY. 

a single one of these layers. It follows from this that the 
path of a raj of light through the atmosphere is not a 
straight line like A C\ but a curve. We may suppose 
this curve to be represented in Fig. 72, where the num- 
ber of layers has been taken very small to avoid confusing 
the drawing. 

Let C be the centre and A a point of the surface of the 
earth ; let S be a star, and <^6 a ray from the star 
which is refracted at the various layers into which we sup- 
pose the atmosphere to be divided, and which finally 




TIG. 72.— REFRACTION OF LAYERS OF AIR. 

enters the *;ye of an observer at A in the apparent direc- 
tion A S'. He will then see the star in the direction £* 
instead of that of S, and S A ■&', the refraction, will 
throw the star nearer to the zenith Z. 

The angle S' A Z is the apparent zenith distance of S ; 
the true zenith distance of /S r is Z A S, and this may be 
assumed to coincide with Se, as for all heavenly bodies 
except the moon it practically does. The line Se pro- 
longed will meet the line A Z in a point above A, sup- 
pose at b'. 



REFRACTION. 



237 



Law of Refraction. — A consideration of the physical condi- 
tions involved has led to the following form for the refraction in 
zenith distance (A £), 

(A0 = ^Uan(£'-3(AC)), 

in which C is the apparent zenith distance of the 'star, and A is a 
constant to be determined by observation. A is found to be about 
57", so that we may write (A Q = 57" tan £' approximately. 

This expression gives what is called the mean refraction — that is, 
the refraction corresponding to a mean state of the barometer and 
thermometer. It is clear that changes in the temperature and pres- 
sure will affect the density of the air, and hence its refractive power. 
The tables of the mean refraction made by Bessel, based on a more 
accurate formula than the one above, are now usually used, and these 
are accompanied by auxiliary tables giving the small corrections for 
the state of the meteorological instruments. 

Let us consider some of the consequences of refraction, and for 
our purpose we may take the formula (A Q = 57" tan £', as it 
very nearly represents the facts. At C — (A £) = 0, or at the 
apparent zenith there is no refraction. This we should have antici- 
pated as the incident ray is itself normal to the refracting surface. 

The following extract from a refraction table gives the amount of 
refraction at various zenith distances : 



c 


(AC) 


V 


(AC) 


0° 


0' 0" 


70° 


2' 39" 


10° 


<y io" 


80° 


5' 20" 


20° 


0' 33" 


85° 


10' 0" 


45° 


0' 58* 


88° 


18' 0" 


50° 


r 09* 


89° 


24' 25" 


60° 


1' 40" 


90° 


34' 30* 



Quantity and Effects of Refraction. — At 45° the refrac- 
tion is about 1', and at 90° it is 34' 30" — that is, bodies at 
the zenith distances of 45° and 90° appear elevated above 
their true places by 1' and 34^-' respectively. If the sun 
has just risen — that is, if its lower limb is just in apparent 
contact with the horizon, it is, in fact, entirely below the 
true horizon, for the refraction (35') has elevated its cen- 
tre by more than its whole apparent diameter (32'). 

The moon is full when it is exactly opposite the sun, 
and therefore were there no atmosphere, moon-rise of a 
full moon and sunset would be simultaneous. In fact, 



238 ASTRONOMY. 

both bodies being elevated by refraction, we see the full 
moon risen before the sun has set. On April 20th, 1837, 
the full moon rose eclipsed before the sun had set. 

We see from the table that the refraction varies com- 
parative];^ little between 0° and 60° of zenith distance, but 
that beyond 80° or 85° its variation is quite rapid. 

The refraction on the two limbs of the sun or moon will 
then be different, and of course greater on the lower limb. 
This will apparently be lifted up toward the upper limb 
more than the upper limb is lifted away from it, and 
hence the sun and moon appear oval in shape when near 
the horizon. For example, if the zenith distance of the 
sun's lower limb is 85°, that of the upper will be about 
84° 28', and the refractions from the tables for these two 
zenith distances differ by 1' ; therefore, the sun will ap- 
pear oval in shape, with axes of 32' and 31' approxi- 
mately. 

Determination of Refraction. — If we know the law according 
to which refraction varies — that is, if we have an accurate formula 
which will give ( A Q in terms of C, we can determine the absolute 
refraction for any one point, and from the law deduce it for any 
other points. Thus knowing the horizontal refraction, or the re- 
fraction in the horizon, we can determine the refraction at other 
known zenith distances. 

We know the time of (theoretical or true) sunrise and sunset by 
the formula of § 7, p. 44, and we may observe the time of apparent 
rising and setting of the sun (or a star). The difference of these 
times gives a means of determining the effect of refraction. 

Or, in the observations for latitude by the method of § 8, p. 47, we 
can measure the apparent polar distances of a circumpolar star at 
its upper and lower culmination. Its polar distances above and 
below pole should be equal ; if there were no refraction they would 
be so, but they really differ by a quantity which it is easy to see is 
the difference of the refractions at lower and upper culminations. 
By choosing suitable circumpolar stars at various polar distances, 
this difference may be determined for all polar distances, and there- 
fore at all zenith distances. 

§ 2. ABERRATION AND THE MOTION OF LIGHT. 

Besides refraction, there is another cause which prevents 
our seeing the celestial bodies exactly in the true direction 
in which they lie from us — namely, the progressive mo- 



ABERRATION. 239 

tion of light. We now know that we see objects only 
by the light which emanates from them and reaches onr 
eyes, and we also know that this light requires time to 
pass over the space which separates us from the object. 
After the ray of light once leaves the object, the latter 
may move away, or even be blotted out of existence, but 
the ray of light will continue on its course. Consequent- 
ly when we look at a star, we do not see the star that now 
is, but the star that was several years ago. If it should be 
annihilated, we should still see it during the years which 
would be required for the last ray of light emitted by it to 
reach us. The velocity of light is so great that in all ob- 
servations of terrestrial objects, our vision may be regarded 
as instantaneous. But in celestial observations the time 
required for the light to reach us is quite appreciable and 
measurable. 

The discovery of the propagation of light is among the 
most remarkable of those made by modern science. The 
fact that light requires time to travel was first learned by 
the observations of the satellites of Jupiter. Owing to 
the great magnitude of this planet, it casts a much longer 
and larger shadow than our earth does, and its inner sat- 
ellite is therefore eclipsed at every revolution. These 
eclipses can be observed from the earth, the satellite van- 
ishing from view as it enters the shadow, and suddenly 
reappearing when it leaves it again. The accuracy with 
which the times of this disappearance and reappearance 
could be observed, and the consequent value of such ob- 
servations for the determination of longitudes, led the 
astronomers of the seventeenth century to make a careful 
study of the motions of these bodies. It was, however, 
necessary to make tables by which the times of the eclipses 
could be predicted. It was found by Koemer that these 
times depended on the distance of Jupiter from the earth. 
If he made his tables agree with observations when the 
earth was nearest Jupiter, it was found that as the earth 
receded from Jupiter in its annual course around the sun. 



240 ASTRONOM1. 

the eclipses were constantly seen later, until, when at its 
greatest distance, the times appeared to be 22 minntes late. 
Roemer saw that it was in the highest degree improbable 
that the actual motions of the satellites should be affected 
with any such inequality ; he therefore propounded the 
bold theory that it took time for light to come from Ju- 
piter to the earth. The extreme differences in the times 
of the eclipse being 22 minutes, he assigned this as the time 
required for light to cross the orbit of the earth, and so 
concluded that it came from the sun to the earth in 11 
minutes. We now know that this estimate was too great, 
and that the true time for this passage is about 8 minutes 
and 18 seconds. 

Discovery of Aberration. — At first this theory of Roe- 
mp:r was not fully accepted by his contemporaries. But 
in the year 1729 the celebrated Bradley, afterward As- 
tronomer Royal of England, discovered a phenomenon of 
an entirely different character, which confirmed the theory. 
He was then engaged in making observations on the star 
y Draconis in order to determine its parallax. The effect 
of parallax would have been to make the declination 
greatest in June and least in December, while in March 
and September the star would occupy an intermediate or 
mean position. But the result was entirely different. 
The declinations of June and December were the same, 
showing no effect of parallax ; but instead of remaining 
constant the rest of the year, the declination was some 40 
seconds greater in September than in March, when the 
effect of parallax would be the same. This showed that 
the direction of the star appeared different, not according 
to the position of the earth, but according to the direction 
of its motion around the sun, the star being apparently 
displaced in this direction. 

It has been said that the explanation of this singular 
anomaly was first suggested to Bradley while sailing on 
the Thames. He noticed that when his boat moved rapid- 
ly at right angles to the true direction of the wind, the 



ABERRATION. 241 

apparent direction of the wind changed toward the point 
whither the boat was going. "When the boat sailed in an 
opposite direction, the apparent direction of the wind sud- 
denly changed in a corresponding way. Here was a phe- 
nomenon very analogous to that which he had observed in 
the stars, the direction from which the wind appeared to 
come corresponding to the direction in which the light 
reached the eye. This direction changed with the mo- 
tion of the observer according to the same law in the two 
cases. He now saw that the apparent displacement of the 
star was due to the motion of the rays of light combined 
with that of the earth in its orbit, the apparent direction 
of the star depending, not upon the absolute direction 
from which the ray comes, but upon the relation of this 
direction to the motion of the observer. 

To show how this is, let A B be the optical axis of a 
telescope, and S a star from which emanates a ray mov- 
ing in the true direction S A B'. 
Perhaps the reader will have a clearer 
conception of the subject if he imag- 
ines A B to be a rod which an ob- 
server at B seeks to point at the star 
/S. It is evident that he will point 
this rod in such a way that the ray 
of light shall run accurately along its 
length. Suppose now that the ob- 
server is moving from B toward B f 
with such a velocity that he moves 

from B to B' during the time re- 

Fig 73 
quired for a ray of light to move from 

A to B\ Suppose also that the ray of light S A reaches 

A at the same time that the end of his rod does. Then 

it is clear that while the rod is moving from the position 

A B to the position A' B', the ray of light will move from 

A to B', and will therefore run accurately along the length 

of the rod. For instance, if b is one third of the way 

from B to B', then the light, at the instant of the rod tak- 




242 ASTRONOMY. 

ing the position b a, will be one third of the way from A 
to B ', and will therefore be accurately on the rod. Con- 
sequently, to the observer, the rod will appear to be point- 
ed at the star. In reality, however, the pointing will not 
be in the true direction of the star, but will deviate from 
it by an angle of which the tangent is the ratio of the 
velocity with which the observer is carried along to the 
velocity of light. This presupposes that the motion of the 
observer is at right angles to that of a ray of light. If 
this is not his direction, we mast resolve his velocity into 
two components, one at right angles to the ray and one 
parallel to it. The latter will not affect the apparent di- 
rection of the star, which will therefore depend entirely 
upon the former. 

Effects of Aberration. — The apparent displacement of 
the heavenly bodies thus produced is called the aberration 
of light. Its effect is to cause each of the fixed stars to 
describe an apparent annual oscillation in a very small or- 
bit. The nature of the displacement may be conceived 
of in the following way : Suppose the earth at any moment, 
in the course of its annual revolution, to be moving to- 
ward a point of the celestial sphere, which we may call P. 
Then a star lying in the direction P or in the opposite di- 
rection will suffer no displacement whatever. A star ly- 
ing in any other direction will be displaced in the direc- 
tion of the point P by an angle proportional to the sine of 
its angular distance from P. At 90° from P the dis- 
placement will be a maximum, and its angular amount 
will be such that its tangent will be equal to the ratio of 
the velocity of the earth to that of light. If A be the 
"aberration" of the star, and P S its angular distance 
from the point P, we shall have, 

tan .4 = —.sin PS. 

v 

v f and v being the respective velocities of light and of the 
earth. 



VELOCITY OF LIGHT. 243 

Now, if the star lies near the pole of the ecliptic, its di- 
rection will always be nearly at right angles to the direc- 
tion in which the earth is moving. A little consideration 
will show that it will seem to describe a circle in conse- 
quence of aberration. If, however, it lies in the plane of 
the earth's orbit, then the various points toward which 
the earth moves in the course of the year all lying in the 
ecliptic, and the star being in this same plane, the appar- 
ent motion will be an oscillation back and forth in this 
plane, and in all other positions the apparent motion will 
be in an ellipse more and more flattened as we approach 
the ecliptic. 

. Velocity of Light. — The amount of aberration can be 
determined in two ways. If we know the time which 
light requires to come from the sun to the earth, a simple 
-calculation will enable us to determine the ratio between 
this velocity and that of the earth in its orbit. For in- 
stance, suppose the time to be 498 seconds ; then light 
will cross the orbit of the earth in 996 seconds. The cir- 
cumference of the orbit being found by multiplying its 
diameter by 3-1416, we thus find that, on the supposition 
we have made, light would move around the circumfer- 
ence of the earth's orbit in 52 minutes and 8 seconds. 
But the earth makes this same circuit in 365J days, and 
the ratio of these two quantities is 10090. The maximum 
displacement of the star by aberration will therefore be the 
angle of which the tangent is yo^o-, and this angle we 
find by trigonometrical calculation to be 20" -44. 

This calculation presupposes that we know how long 
light requires to come from the sun. This is not known 
with great accuracy owing to the unavoidable errors with 
which the observations of Jupiter 's satellites are affected. 
It is therefore more usual to reverse the process and de- 
termine the displacement of the stars by direct observa- 
tion, and then, by a calculation the reverse of that we 
have just made, to determine the time required by light 
to reach us from the sun. Many painstaking determina- 



244 ASTRONOMY. 

tions of this quantity have been made since the time of 
Bradley, and as the result of them we may say that the 
value of the ' i constant of aberration, " as it is called, is 
certainly between 20" • 4 and 20" • 5 ; the chances are that it 
does not deviate from 20". 44 by more than two or three 
hundredths of a second. 

It will be noticed that by determining the constant of 
aberration, or by observing the eclipses of the satellites of 
Jupiter, we may infer the time required for light to pass 
from the sun to the earth. But we cannot thus determine 
the velocity of light unless we know how far the sun is. 
The connection between this velocity and the distance of 
the sun is such that knowing one we can infer the other. 
Let us assume, for instance, that the time required for 
light to reach us from the sun is 498 seconds, a time which 
is probably accurate within a single second. Then know- 
ing the distance of the sun, we may obtain the velocity of 
light by dividing it by 498. But, on the other hand, if we 
can determine how many miles light moves in a second, we 
can thence infer the distance of the sun by multiplying it 
by the same factor. During the last century the distance 
of the sun was found to be certainly between 90 and 100 
millions of miles. It was therefore correctly concluded 
that the velocity of light was something less than 200,000 
miles per second, and probably between 180,000 and 
200,000. This velocity has since been determined more 
exactly by the direct measurements at the surface of the 
earth already mentioned 



CHAPTER XI. 

CHRONOLOGY. 
£ 1. ASTRONOMICAL MEASURES OF TIME. 

The most intimate relation of astronomy to the daily 
life of mankind has always arisen from its affording the 
only reliable and accurate measure of long intervals of time. 
The fundamental units of time in all ages have been the 
day, the month, and the year, the first being measured by 
the revolution of the earth on its axis, the second, prim- 
itively, by that of the moon around the earth, and the third 
by that of the earth round the sun. Had the natural month 
consisted of an exact entire number of days, and the year 
of an exact entire number of months, there would have 
been no history of the calendar to write. There being no 
such exact relations, innumerable devices have been tried 
for smoothing off the difficulties thus arising, the mere 
description of which would fill a volume. We shall en- 
deavor to give the reader an idea of the general character 
of these devices, including those from which our own cal- 
endar originated, without wearying him by the introduc- 
tion of tedious details. 

Of the three units of time just mentioned, the most nat- 
ural and striking is the shortest — namely, the day. Mark- 
ing as it does the regular alternations of wakefulness and 
rest for both man and animals, no astronomical observa- 
tions were necessary to its recognition. It is so nearly 
uniform in length that the most refined astronomical obser- 
vations of modern times have never certainly indicated 



246 ASTRONOMY. 

any change. This unif ormity, and its entire freedom from 
all ambiguity of meaning, have always made the day a 
common fundamental unit of astronomers. Except for 
the inconvenience of keeping count of the great number 
of days between remote epochs, no greater unit would 
ever have been necessary, and we might all date our let- 
ters by the number of days after Christ, or after a sup- 
posed epoch of creation. 

The difficulty of remembering great numbers is such 
that a longer unit is absolutely necessary, even in keeping 
the reckoning of time for a single generation. Such a 
unit is the year. The regular changes of seasons in all ex- 
tra-tropical latitudes renders this unit second only to the 
day in the prominence with which it must have struck the 
minds of primitive man. These changes are, however, so 
slow and ill-marked in their progress, that it would have 
been scarcely possible to make an accurate determination 
of the length of the year from the observation of the sea- 
sons. Here astronomical observations came to the aid of 
our progenitors, and, before the beginning of extant his- 
tory, it was known that the alternation of seasons was due 
to the varying declination of the sun, as the latter seemed 
to perform its annual course among the stars in the 
" oblique circle" or ecliptic. The common people, who did 
not understand the theory of the sun's motion, knew that 
certain seasons were marked by the position of certain 
bright stars relatively to the sun — that is, by those stars 
rising or setting in the morning or evening twilight. 
Thus arose two methods of measuring the length of the 
year — the one by the time when the sun crossed the equi- 
noxes or solstices, the other when it seemed to pass a cer- 
tain point among the stars. As we have already explain- 
ed, these years were slightly different, owing to the pre- 
cession of the equinoxes, the first or equinoctial year being 
a little less and the second or sidereal year a little greater 
than 365£ days. 

The number of days in a year is too great to admit of 



CHRONOLOGY. 247 

their being easily remembered without any break ; an in- 
termediate period is therefore necessary. Such a period 
is measured by the revolution of the moon around the 
earth, or, more exactly, by the recurrence of new moon, 
which takes place, on the average, at the end of nearly 
29^- days. The nearest round number to this is 30 days, 
and 12 periods of 30 days each only lack 5J days of being 
a year. It has therefore been common to consider a year 
as made up of 12 months, the lack of exact correspondence 
being filled by various alterations of the length of the 
month or of the year, or by adding surplus days to each 
year. 

The true lengths of the day, the month, and the year 
having no common divisor, a difficulty arises in attempting 
to make months or days into years, or days into months, 
owing to the fractions which will always be left over. At 
the same time, some rule bearing on the subject is necessary 
in order that people may be able to remember the year, 
month, and day. Such rules are found by choosing some 
cycle or period which is very nearly an exact number of 
two units, of months and of days for example, and by di- 
viding this cycle up as evenly as possible. The principle 
on which this is done can be seen at once by an example, 
for which we shall choose the lunar month. The true 
length of this month is 29 • 5305884 days. We see that 
two of these months is only a little over 59 days ; so, if 
we take a cycle of 59 days, and divide it into two months, 
the one of 30 and the other of 29 days, we shall have a 
first approximation to a true average month. But our 
cycle will be too short by d -061, the excess of two months 
over 59 days, and this error will be added at the end of 
every cycle, and thus go on increasing as long as the cycle 
is used without change. At the end of 16 cycles, or of 
32 lunar months, the accumulated error will amount to 
one day. At the end of this time, if not sooner, we 
should have to add a day to one of the months. 

Seeing that we shall ultimately be wrong if we have a 



248 ASTR0N03IT. 

two-month cycle, we seek for a more exact one. Each 
month of 30 days is nearly d • 47 too long, and each month 
of 29 days is rather more than d • 53 too short. So in the 
long run the months of 30 days ought to be more numer- 
ous than those of 29 days in the ratio that 53 bears to 
47, or, more exactly, in the ratio that • 5305884 bears to 
• 4694116. A close approximation will be had by having 
the long months one eighth more numerous than the short 
ones, the numbers in question being nearly in the ratio of 
9:8. So, if we take a cycle of 17 months, 9 long and 8 
short ones, we find that 9 x 30 + 8 x 29 = 502 days for 
the assumed length of our cycle, whereas the true length 
of 17 months is very near 502 d • 0200. The error will 
therefore be • 02 of a day for every cycle, and will not 
amount to a day till the end of 50 cycles, or nearly 70 
years. 

A still nearer approach will be found by taking a cycle 
of 49 months, 26 to be long and 23 short ones. These 
49 months will be composed of 26 x 30 + 23 x 29 = 
1447 days, whereas 49 true lunar months will comprise 
1446 • 998832 days. Each cycle will therefore be too long 
by only -001168 of a day, and the error would not amount 
to a day till the end of 84 cycles, or more than 3000 years. 

Although these cycles are so near the truth, they could 
not be used with convenience because they would begin 
at different times of the year. The problem is therefore 
to find a cycle which shall comprise an entire number of 
years. We shall see hereafter what solutions of this 
problem were actually found. 

§ 2. FORMATION OP CALENDARS. 

The months now or heretofore in use among the peoples 
of the globe may for the most part be divided into two 
classes : 

(1.) The lunar month pure and simple, or the mean 
interval between successive new moons. 



THE CALENDAR. 249 

(2.) An approximation to the twelfth part of a year, 
without respect to the motion of the moon. 

The Lunar Month. — The mean interval between con- 
secutive new moons being nearly 29£ days, it was common 
in the use of the pure lunar month to have months of 29 and 
30 days alternately. This supposed period, however, as just 
shown, will fall short by a day in about 2^- years. This de- 
fect was remedied by introducing cycles containing rather 
more months of 30 than of 29 days, the small excess of 
long months being spread uniformly through the cycle. 
Thus the Greeks had a cycle of 235 months (to be soon 
described more fully), of which 125 were full or long 
months, and 110 were short or deficient ones. We see 
that the length of this cycle was 6940 days (125 x 30 + 
110 x 29), whereas the length of 235 true lunar months 
is 235 x 29 • 53058 = 6939 . 688 days. The cycle was there- 
fore too long by less than one third of a day, and the error 
of count would amount to only one day in more than TO 
years. The Mohammedans, again, took a cycle of 360 
months, which they divided into 169 short and 191 long 
ones. The length of this cycle was 10631 days, while the 
true length of 360 lunar months is 10631-012 days. The 
count would therefore not be a day in error until the end of 
about 80 cycles, or nearly 23 centuries. This month there- 
fore follows the moon closely enough for all practical pur- 
poses. 

Months other than Lunar. — The complications of the 
system just described, and the consequent difficulty of 
making the calendar month represent the course of the 
moon, are so great that the pure lunar month was gen- 
erally abandoned, except among people whose religion re- 
quired important ceremonies at the time of new moon. 
In cases of such abandonment, the } T ear has been usually 
divided into 12 months of slightly different lengths. The 
ancient Egyptians, however, had 12 months of 30 days 
each, to which they added 5 supplementary days at the 
close of each year. 



250 ASTRONOMY. 

Kinds of Year. — As we find two different systems of 
months to have been used, so we may divide the calendar 
years into three classes — namely : 

(1.) The lunar year, of 12 lunar months. 

(2.) The solar year. 

(3.) The combined luni-solar year. 

The Lunar Year. — We have already called attention to 
the fact that the time of recurrence of the year is not well 
marked except by astronomical phenomena which the 
casual observer would hardly remark. But the time of 
new moon, or of beginning of the month, is always well 
marked. Consequently, it was very natural for people to 
begin by considering the year as made up of twelve luna- 
tions, the error of eleven days being unnoticeable in a 
single year, unless careful astronomical observations were 
made. Even when this error was fully recognized , it might 
be considered better to use the regular year of 12 lunar 
months than to use one of an irregular or varying number 
of months. Such a year is the religious one of the Mo- 
hammedans to this day. The excess of 11 days will 
amount to a whole year in 33 years, 32 solar years being 
nearly equal to 33 lunar years. In this period therefore 
each season will have coursed through all times of the 
year. The lunar year has therefore been called the 
' ' wandering year. ' ' 

The Solar Year. — In forming this year, the attempt to 
measure the year by revolutions of the moon is entirely 
abandoned, and its length is made to depend entirely on 
the change of the seasons. The solar year thus indicated 
is that most used in both ancient and modern times. Its 
length has been known to be nearly 3 65 J days from the 
times of the earliest astronomers, and the system adopted 
in our calendar of having three years of 365 days each, fol- 
lowed by one of 366 days, has been employed in China 
from the remotest historic times. This year of 365^ days 
is now called by us the Julian Year, after Julius Caesar, 
from whom we obtained it 



THE CALENDAR. 251 

The Luni-Solar Year. — If the lunar months must, in 
some way, be made up into solar years of the proper av- 
erage length, then these years must be of unequal length, 
some having twelve months and others thirteen. Thus, a 
period or cycle of eight years might be made up of 99 
lunar months, 5 of the years having 12 months each, and 
3 of them 13 months each. Such a period would comprise 
2923J days, so that the average length of the year would 
be 365 days 10^ hours. This is too great by about 4 hours 
42 minutes. This very plan was proposed in ancient 
Greece, but it was superseded by the discovery of the 
Metonio Cycle, which figures in our church calendar to 
this day. A luni-solar year of this general character was 
also used by the Jews. 

The Metonic Cycle. — The preliminary considerations we 
have set forth will now enable us to understand the origin 
of our own calendar. We begin with the Metonic Cycle 
of the ancient Greeks, which still regulates some religious 
festivals, although it has disappeared from our civil reck- 
oning of time. The necessity of employing lunar months 
caused the Greeks great difficulty in regulating their cal- 
endar so as to accord with their rules for religious feasts, 
until a solution of the problem was found by Meton, about 
433 B.C. The great discovery of Meton w T as that a period 
or cycle of 6940 days could be divided up into 235 lunar 
months, and also into 19 solar years. Of these months, 
125 were to be of 30 days each, and 110 of 29 days each, 
which would, in all, make up the required 6940 days. To 
see how nearly this rule represents the actual motions of 
the sun and moon, we remark that : 

Days. Hours. Min. 

235 lunations require 6939 16 31 

19 Julian years " 6939 IS 

19 true solar years require 6939 14 27 

We see that though the cycle of 6940 days is a few hours 
too long, yet, if we take 235 true lunar months, we iind 



252 ASTRONOMY. 

their whole duration to be a little less than 19 Julian years of 
365 \ days each, and a little more than 19 true solar years. 

The problem now was to take these 235 months and divide 
them up into 19 years, of which 12 should have 12 months 
each, and 7 should have 13 months each. The long years, 
or those of 13 months, were probably those corresponding 
to the numbers 3, 5, 8, 11, 13, 16, and 19, while the first, 
second, fourth, sixth, etc., were short years. In general, 
the months had 29 and 30 days alternately, but it was 
necessary to substitute a long month for a short one every 
two or three years, so that in the cycle there should be 
125 long and 110 short months. 

Golden Number. — This is simply the number of the 
year in the Metonic Cycle, and is said to owe its appella- 
tion to the enthusiasm of the Greeks over Meton's dis- 
covery, the authorities having ordered the division and 
numbering of the years in the new calendar to be in- 
scribed on public monuments in letters of gold. The rule 
for finding the golden number is to divide the number of 
the year by 19, and add 1 to the remainder. From 1881 
to 1899 it may be found by simply subtracting 1880 from 
the year. It is employed in our church calendar for find- 
ing the time of Easter Sunday. 

Period of Callypus. — We have seen that the cycle of 
6940 days is a few hours too long either for 235 lunar 
months or for 19 solar years. Callypus therefore sought 
to improve it by taking one day off of every fourth cycle, 
so that the four cycles should have 27759 days, which 
were to be divided into 940 months and into 76 years. 
These years would then be Julian years, while the recur- 
rence of new moon would only be six hours in error at the 
end of the 76 years. Had he taken a day from every 
third cycle, and from some year and month of that cycle, 
he would have been yet nearer the truth. 

The Mohammedan Calendar. — Among the most remark- 
able calendars which have remained in use to the present 
time is that of the Mohammedans. The year is composed 



TEE MOHAMMEDAN CALENDAR. 253 

of 12 lunar months, and therefore, as already mentioned, 
does not correspond to the course of the seasons. As with 
other systems, the problem is to find such a cycle that an 
entire number of these lunar years shall correspond to an 
integral number of days. Multiplying the length of the 
lunar month by 12, we find the true length of the lunar 
year to be 354 • 36706 days. The fraction of a day being 
not far from one third, a three-year cycle, comprising two 
years of 354 and one of 355 days, would be a first approx- 
imation to three lunar years, but would still be one tenth 
of a day too short. In ten such cycles or thirty years, 
this deficiency would amount to an entire day, and by add- 
ing the day at the end of each tenth three-year cycle, 
a very near approach to the true motion of the moon 
will be obtained. This thirty-year cycle will consist of 
10631 days, while the true length of 360 lunar months is 
10631 -0116 days. The error will not amount to a day until 
the end of 87 cycles, or 2610 years, so that this system is 
accurate enough for all practical purposes. The common 
Mohammedan year of 354 days is composed of months 
containing alternately 30 and 29 days, the first having 
30 and the last 29. In the years of 355 days the alter- 
nation is the same, except that one day is added to the last 
month of the year. 

The old custom was to take for the first day of the 
month that following the evening on which the new moon 
could first be seen in the west. It is said that before the 
exact arrangement of the Mohammedan calendar had been 
completed, the rule was that the visibility of the crescent 
moon should be certified by the testimony of two wit- 
nesses. The time of new moon given in our modern 
almanacs is that when the moon passes nearly between us 
and the sun, and is therefore entirely invisible. The moon 
is generally one or two days old before it can be seen in the 
evening, and, in consequence, the lunar month of the Mo- 
hammedans and of others commences about two days after 
the actual almanac time of new moon. 



254 A8TB0N0MT. 

The civil calendar now in use throughout Christendom 
had its origin among the Romans, and its foundation was 
laid by Julius Caesar. Before his time, Rome can hardly be 
said to have had a chronological system, the length of the 
year not being prescribed by any invariable rule, and be- 
ing therefore changed from time to time to suit the caprice 
or to compass the ends of the rulers. Instances of this 
tampering disposition are familiar to the historical student. 
It is said, for instance, that the Gauls having to pay a 
certain monthly tribute to the Romans, one of the govern- 
ors ordered the year to be divided into 14 months, in 
order that the pay days might recur more frequently. To 
remedy this, Cjssar called in the aid of Sosigenes, an as- 
tronomer of the Alexandrian school, and by them it was 
arranged that the year should consist of 365 days, with the 
addition of one day to every fourth year. The old Roman 
months were afterward adjusted to the Julian year in 
such a way as to give rise to the somewhat irregular 
arrangement of months which we now have. 

Old and New Styles. — The mean length of the Julian 
year is 365^ days, about 11J minutes greater than that of 
the true equinoctial year, which measures the recurrence 
of the seasons. This difference is of little practical im- 
portance, as it only amounts to a week in a thousand years, 
and a change of this amount in that period is productive 
of no inconvenience. But, desirous to have the year as 
correct as possible, two changes were introduced into the 
calendar by Pope Gregory XIII. with this object. They 
were as follows : 

1. The day following October 4, 1582, was called the 
15th instead of the 5th, thus advancing the count 10 days. 

2. The closing year of each century, 1600, 1700, etc., 
instead of being always a leap year, as in the Julian 
calendar, is such only when the number of the century is 
divisible by 4. Thus while 1600 remained a leap year, as 
before, 1700, 1800, and 1900 were to be common years. 

This change in the calendar was speedily adopted by all 



THE CALENDAR. 255 

Catholic countries, and more slowly by Protestant ones, 
England holding out until 1752. In Russia it has never 
been adopted at all, the Julian calendar being still con- 
tinued without change. ' The Russian reckoning is there- 
fore 12 days behind ours, the ten days dropped in 1582 
being increased by the days dropped from the years 1700 
and 1800 in the new reckoning. This modified calendar 
is called the Gregorian Calendar, or New Style, while the 
old system is called the Julian Calendar, or Old Style. 

It is to be remarked that the practice of commencing 
the year on January 1st was not universal until compara- 
tively recent times. During the first sixteen centuries of 
the Julian calendar there was such an absence of definite 
rules on this subject, and such a variety of practice on the 
part of different powers, that the simple enumeration of 
the times chosen by various governments and pontiffs for 
the commencement of the year would make a tedious 
chapter. The most common times of commencing were, 
perhaps, March 1st and March 22d, the latter being the 
time of the vernal equinox. But January 1st gradually 
made its way, and became universal after its adoption by 
England in 1752. 

Solar Cycle and Dominical Letter. — In our church cal- 
endars January 1st is marked by the letter A, January 2d 
by B, and so on to G, when the seven letters begin over 
again, and are repeated through the year in the same 
order. Each letter there indicates the same day of the 
week throughout each separate year, A indicating the day 
on which January 1st falls, B the day following, and so 
on. An exception occurs in leap years, when February 
29th and March 1st are marked by the same letter, so that 
a change occurs at the beginning of March. The letter 
corresponding to Sunday on this scheme is called the Do- 
minical or Sunday letter, and, when we once know what 
letter it is, all the Sundays of the year are indicated by 
that letter, and hence all the other days of the week by 
their letters. In leap years there will be two Dominical 



256 ASTRONOMY. 

letters, that for the last ten months of the year being the 
one next preceding the letter for January and February. 
In the Julian calendar the Dominical letter must always 
recur at the end of 28 years (besides three recurrences at 
unequal intervals in the mean time). This period is called 
the solar cycle, and determines the clays of the week on 
which the days of the month fall during each year. 

Since any day of any year occurs one day later in the 
week than it did the year before, or two days later when 
a 29th of February has intervened, the Dominical letters 
recur in the order G, F, E, D, C, B, A, G, etc. This may 
also be expressed by saying that any day of a past year 
occurred one day earlier in the week for every year that 
has elapsed, and, in addition, one day earlier for every 
29th of February that has intervened. This fact will make 
it easy to calculate the day of the week on which any his- 
torical event happened from the day corresponding in any 
past or future year. Let us take the following example : 

On what day of the week was Washington born, the 
date being 1732, February 22d, knowing that February 
22d, 1879, fell on Saturday. The interval is 147 years : 
dividing by 4 we have a quotient of 36 and a remainder 
of 3, showing that, had every fourth year in the interval 
been a leap year, there were either 36 or 37 leap years. 
As a February 29th followed only a week after the date, 
the number must be 37 ;* but as 1800 was dropped from 
the list of leap years, the number was really only 36. 
Then 147 + 36 = 183 days advanced in the week. Di- 
viding by 7, because the same clay of the week recurs 
after seven clays, we find a remainder of 1. So February 
22d, 1879, is one day further advanced than was February 
22d, 1732 ; so the former being Saturday, Washington 
was born on Friday. 

* Perhaps the most convenient way of deciding whether the remainder 
does or does not indicate an additional leap year is to subtract it from the 
last date, and see whether a February 29th then intervenes. Subtract- 
ing 3 years from February 22d, 1879, we have February 22d, 1876, 
and a 29th occurs between the two dates, only a week after the first. 



DIVISION OF THE DAT 257 

§ 3. DIVISION OP THE DAY. 

The division of the day into hours was, in ancient and 
mediaeval times, effected in a way very different from that 
which we practice. Artificial time-keepers not being in 
general use, the two fundamental moments were sunrise 
and sunset, which marked the day as distinct from the 
night. The first subdivision of this interval was marked 
by the instant of noon, when the sun was on the meridian. 
The day was thus subdivided into two parts. The night 
was similarly divided by the times of rising and culmina- 
tion of the various constellations. Euripides (480-407 
b.c.) makes the chorus in Rhesus ask : 

" Chorus. — Whose is the guard ? Who takes my turn? The first 
signs are setting, and the seven Pleiades are in the sky, and the Eagle glides 
midway through heaven. Awake ! Why do you delay ? Awake from 
your beds to watch ! See ye not the brilliancy of the moon ? • Morn, 
morn indeed is approaching, and hither is one of the forerunning stains." 
— The Tragedies of Euripides. Literally Translated by T. A. Buckley. 
London : H. G. Bohn. 1854. Vol. 2, p. 322. 

The interval between sunrise and sunset was divided 
into twelve equal parts called hours, and as this interval 
varied with the season, the length of the hour varied also. 
The night, whether long or short, was divided into hours 
of the same character, only, when the night hours were 
long, those of the day were short, and vice versa. These 
variable hours were called temporary hours. At the time 
of the equinoxes, both the day and the night hours were 
of the same length with those we. use — namely, the twenty- 
fourth part of the day ; these were therefore called equi- 
noctial hours. 

The use of these temporary hours was intimately as- 
sociated with the time of beginning of the day. Instead 
of commencing the civil day at midnight, as we do, it was 
customary to commence it at sunset. The Jewish Sabbath, 
for instance, commenced as soon as the sun set on Friday, 
and ended when it set on Saturday. This made a more 
distinctive division of the astronomical day than that 



258 ASTRONOMY. 

which we employ, and led naturally to considering the 
day and the night as two distinct periods, each to be di- 
vided into 12 hours. 

So long as temporary hours were used, the beginning of 
the day and the beginning of the night, or, as we should 
call it, six o'clock in the morning and six o'clock in the 
evening, were marked by the rising and setting of the sun ; 
but when equinoctial hours were introduced, neither sun- 
rise nor sunset could be taken to count from, because both 
varied too much in the course of the year. It therefore 
became customary to count from noon, or the time at 
which the sun passed the meridian. The old custom of 
dividing the day and the night each into 12 parts was con- 
tinued, the first 12 being reckoned from midnight to 
noon, and the second from noon to midnight. The day 
was made to commence at midnight rather than at noon 
for obvious reasons of convenience, although noon was of 
course the point at which the time had to be determined. 

Equation of Time. — To any one who studied the annual 
motion of the sun, it must have been quite evident that 
the intervals between its successive passages over the 
meridian, or between one noon and the next, could not 
be the same throughout the year, because the apparent 
motion of the sun in right ascension is not constant. It 
will be remembered that the apparent revolution of the 
starry sphere, or, which is the same thing, the diurnal 
revolution of the earth upon its axis, may be regarded 
as absolutely constant for all practical purposes. This rev- 
olution is measured around in right ascension as explained 
in the opening chapter of this work. If the sun increased 
its right ascension by the same amount every day, it would 
pass the meridian 3 m 56 s later every day, as measured by 
sidereal time, and hence the intervals between successive 
passages would be equal. But the motion of the sun in 
right ascension is unequal from two causes : (1) the un- 
equal motion of the earth in its annual revolution around 
it, arising from the eccentricity of the orbit, and (2) the 



APPARENT AND MEAN TIME. 259 

obliquity of the ecliptic. How the first cause produces an 
inequality is obvious, and its approximate amount is readily 
computed. We have seen that the angular velocity of a 
planet around the sun is inversely as the square of its ra- 
dius vector. Taking the distance of the earth from the sun 
as unity, and putting e for the eccentricity of its orbit, its 
greatest distance about the end of June is 1 + e = 1-0168, 
and its least distance about the end of December is 
1 — 0- 0168. The squares of these quantities are 1 • 034 and 
1 — 034 very nearly ; therefore the motion is about one 
thirtieth greater than the mean in December and one 
thirtieth less in June. The mean motion is 3 in 56 3 ; the 
actual motion therefore varies from 3 nl 48 s to 4 m 4 s . 

The effect of the obliquity of the ecliptic is still greater. 
When the sun is near the equinox, its motion along the 
ecliptic makes an angle of 23^° with the parallels of dec- 
lination. Since its motion in right ascension is reckoned 
along the parallel of declination, we see that it is equal to 
the motion in longitude multiplied by the cosine of 23J°. 
This cosine is less than unity by about 0.08 ; therefore 
at the times of the equinox the mean motion is diminished 
by this fraction, or by 20 seconds. Therefore the days 
are then 20 seconds shorter than they would be were there 
no obliquity. At the solstices the opposite effect is pro- 
duced. Here the different meridians of right ascension 
are nearer together than they are at the equator in the 
proportion of the cosine of 23^° to unity ; therefore, when 
the sun moves through one degree along the ecliptic, it 
changes its right ascension by 1-08° ; here, therefore, the 
days are about 19 seconds longer than they would be if the 
obliquity of the ecliptic was zero. 

We thus have to recognize two slightly different kinds 
of days : solar days and mean days. A solar day is the 
interval of time between two successive transits of the sun 
over the same meridian, while a mean day is the mean of 
all the solar days in a year. If we had two clocks, the 
one going with perfect uniformity, but regulated so as to 



260 ASTRONOMY. 

keep as near the sun as possible, and the other changing 
its rate so as to always follow the sun, the latter would gain 
or lose on the former by amounts sometimes rising to 22 
seconds in a day. The accumulation of these variations 
through a period of several months would lead to such 
deviations that the sun-clock would be 14 minutes slower 
than the other during the first half of February, and 16 
minutes faster during the first week in November. The 
time-keepers formerly nsed were so imperfect that these 
inequalities in the solar day were nearly lost in the neces- 
sary irregularities of the rate of the clock. All clocks 
were therefore set by the sun as often as was found neces- 
sary or convenient. But during the last century it was 
found by astronomers that the use of units of time vary- 
ing in this way led to much inconvenience ; they there- 
fore substituted mean time for solar or apparent time. 

Mean time is so measured that the hours and days shall 
always be of the same length, and shall, on the average, be 
as much behind the sun as ahead of it. We may imagine 
a fictitious or mean sun moving along the equator at the 
rate of 3 m 56 s in right ascension every day. Mean time 
will then be measured by the passage of this fictitious sun 
across the meridian. Apparent time was used in ordinary 
life after it was given up by astronomers, because it was 
very easy to set a clock from time to time as the sun 
passed a noon-mark. But when the clock was so far im- 
proved that it kept much better time than the sun did, it 
was found troublesome to keep putting it backward and 
forward, so as to agree with the sun. Thus mean time 
was gradually introduced for all the purposes of ordinary 
life except in very remote country districts, where the 
farmers may find it more troublesome to allow for an equa- 
tion of time than to set their clocks by the sun every few 
days. 

The common household almanac should give the equa- 
tion of time, or the mean time at which the sun passes the 
meridian, on each day of the year. Then, if any one wishes 



IMPROVING THE CALENDAR. 261 

to set his clock, he knows the moment of the sun passing 
the meridian, or being at some noon-mark, and sets his 
time-piece accordingly. For all purposes where accurate 
time is required, recourse must be had to astronomical ob- 
servation. It is now customary to send time-signals every 
day at noon, or some other hour agreed upon, from obser- 
vatories along the principal lines of telegraph. Thus at 
the present time the moment of Washington noon is sig- 
nalled to New York, and over the principal lines of rail- 
way to the South and West. Each person within reach of 
a telegraph-office can then determine his local time by cor- 
recting these signals for the difference of longitude. 

§ 4. REMARKS ON IMPROVING- THE CALENDAR. 

It is an interesting question whether our calendar, this 
product of the growth of ages, which we have so rapidly 
described, would admit of decided improvement if we 
were free to make a new one with the improved materials 
of modern science. This question is not to be hastily an- 
swered in the affirmative. Two small improvements are 
undoubtedly practicable : (1) a more regular division of 
the 365 days among the months, giving February 30 days, 
and so having months of 30 and 31 days only ; (2) putting 
the additional clay of leap year at the end of the year in- 
stead of at the end of February. The smallest change 
from our present system would be made by taking the two 
additional days for February, the one from the end of 
July, and the other from the end of December, leaving 
the last with 30 days in common years and 31 in leap 
years. When we consider more radical changes than this, 
we find advantages set off by disadvantages. For in- 
stance, it would on some accounts be very convenient to 
divide the year into 13 months of 4 weeks each, the last 
month having one or two extra days. The months would 
then begin on the same day of the week through each 
year, and would admit of a much more convenient subdi- 



262 ASTRONOMY. 

vision into halves and quarters than they do now. But the 
year would not admit of such a subdivision without divid- 
ing the months also, and it is possible that this inconven- 
ience would balance the conveniences of the plan. 

An actual attempt in modern times to form an entirely 
new calendar is of sufficient historic interest to be men- 
tioned in this connection. We refer to the so-called Repub- 
lican Calendar of revolutionary France. The year some- 
times had 365 and sometimes 366 days, but instead of 
having the leap years at defined intervals, one was inserted 
whenever it might be necessary to make the autumnal 
equinox fall on the first day of the year. The division of 
the year was effected after the plan of the ancient Egyp- 
tians, there being 12 months of 30 days each, followed by 
5 or 6 supplementary days to complete the year, which 
were kept as feast-days.* The sixth day of course occur- 
red only in the leap years, or Franciads as they were call- 
ed. It was called the Day of the Revolution, and was set 
apart for a quadrennial oath to remain free or die. 

No attempt was made to fit the new calendar to the old 
one, or to render the change natural or convenient. The 
year began with the autumnal equinox, or September 22d 
of the Gregorian calendar ; entirely new names were 
given to the months ; the week was abolished, and in lieu 
of it the month was divided into three decades, the last or 
tenth day of each decade being a holiday set apart for the 
adoration of some sentiment. Even the division of the day 
into 24 hours was done away with, and a division into 
ten hours was substituted. 

The Republican Calendar was formed in 1793, the year 
1 commencing on September 22d, 1792, and it was 
abolished on January 1st, 1806, after 13 years of con- 
fusion. 

* They received the nickname of sans-culottides, from the opponents 
of the new state of things. 



THE ASTRONOMICAL EPHEMERI8. 263 

% 5. THE ASTRONOMICAL EPHEMERIS, OR NAU- 
TICAL ALMANAC. 

The Astronomical Ephemeris, or, as it is more com- 
monly called, the Nautical Almanac, is a work in which 
celestial phenomena and the positions of the heavenly 
bodies are computed in advance. The need of such a work 
must have been felt by navigators and astronomers from 
the time that astronomical predictions became sufficiently 
accurate to enable them to determine their position on the 
surface of the earth. At first works of this class were pre- 
pared and published by individual astronomers who had 
the taste and leisure for this kind of labor. Manfeedi, 
of Bonn, published Ephemerides in two volumes, which 
gave the principal aspects of the heavens, the positions of 
the stars, planets, etc., from 1715 until 1725. This work 
included maps of the civilized world, showing the paths of 
the principal eclipses during this interval. 

The usefulness of such a work, especially to the naviga- 
tor, depends upon its regular appearance on a uniform plan 
and upon the fulness and accuracy of its data ; it was there- 
fore necessary that its issue should be taken up as a gov- 
ernment work. Of works of this class still issued the 
earliest was the Connaissance dcs Temps of France, the 
first volume of which was published by Picard in 1679, 
and which has been continued without interruption until 
the present time. The publication of the British Nautical 
Almanac was commenced in the year 1767 on the repre- 
sentations of the Astronomer Royal showing that such a 
work would enable the navigator to determine his longi- 
tude within one degree by observations of the moon. An 
astronomical or nautical almanac is now published annually 
by each of the governments of Germany, Spain, Portugal, 
France, Great Britain, and the United States. They have 
gradually increased in size and extent with the advancing 
wants of the astronomer until those of Great Britain and 
this country have become octavo volumes of between 500 



264 ASTRONOMY. 

and 600 pages. These two are published three years or 
more beforehand, in order that navigators going on long 
voyages may supply themselves in advance. The Ameri- 
can Ephemeris and Nautical Almanac lias been regular- 
ly published since 1855, the first volume being for that 
year. It is designed for the use of navigators the world 
over, and the greater part of it is especially arranged for 
the use of astronomers in the United States. 

The immediate object of publications of this class is to 
enable the wayfarer and traveller upon land and the voy- 
ager upon the ocean to determine their positions by obser- 
vations of the heavenly bodies. Astronomical instruments 
and methods of calculation have been brought to such a 
degree of perfection that an astronomer, armed with a nau- 
tical almanac, a chronometer regulated to Greenwich or 
Washington time, a catalogue of stars, and the necessary 
instruments of observation, can determine his position at 
any point on the earth's surface within a hundred yards 
by a single night's observations. If his chronometer is. 
not so regulated, he can still determine his latitude, but not 
his longitude. He could, however, obtain a rough idea 
of the latter by observations upon the planets, and come 
within a very few miles of it by a single observation on 
the moon. 

The Ephemeris furnishes the fundamental data from 
which all our household almanacs are calculated. 

The principal quantities given in the American Ephemeris for 
each year are as follows : 

The positions, of the sun and the principal large planets for Green- 
wich noon of every day in each year. 

The right ascension and declination of the moon's centre for 
every hour in the year. 

The distance of the moon from certain bright stars and planets 
for every third hour of the year. 

The right ascensions and declinations of upward of two hundred 
of the brighter fixed stars, corrected for precession, nutation, and 
aberration, for every ten days. 

The positions of the principal planets at every visible transit over 
the meridian of Washington. 

Compete elements of all the eclipses of the sun and moon, with 



THE EPHEMERIS. 



265 



maps showing the passage of the moon's shadow or penumbra over 
those regions of the earth where the eclipses will be visible, and 
tables whereby the phases of the eclipses can be accurately com- 
puted for any place. 

Tables for predicting the occultations of stars by the moon. 

Eclipses of Jupiter's satellites and miscellaneous phenomena. 

To give the reader a still further idea of the Ephemeris, we pre- 
sent a small portion of one of its pages for the year 1882 : 



February, 1882 — at Greenwich Mean Noon. 



Day of 

the 
week. 


o 

cm *e 

ft 






The 


Sun's 




Equation 
of time to 
be sub- 
tracted 
from m ean 
time. 


U 

o 
A 


Sidere 
or rig 
censic 
mean 


al time 


Apparent 
right ascen- 
sion. 


Diff. 
for 1 

hour. 


Apparent de- 
clination. 


Diff. 
fori 
hour. 


it as- 
>n of 
sun. 


Wed. 
Thur. 
Frid. 


1 

2 
3 


ii. 
21 

21 
21 


>r. 

4 
8 


s. 
13-04 
16-84 
19-82 


s. 
10-175 
10-141 
10-107 


o 

S17 
16 
16 


2 

45 

27 


22-4 

5-4 

30-9 


+42-82 
43-57 
44-30 


M. s. 
13 51-34 

13 58-58 

14 5-01 


s. 
0-318 
0-284 
0-250 


ii. 
20 
£0 

20 


M. 

46 
50 

54 


s. 
21-70 
18-26 
14-81 


Sat. 
■Sun. 
Mon. 


4 
5 
6 


21 
21 
21 


12 

16 

20 


21-98 
23-33 

23-88 


10 073 
10-040 
10-007 


16 

15 
15 


9 

51 
33 


39-2 

30-8 

6-1 


+44-99 
45-69 
46-36 


14 10-61 
14 15-41 
14 19-40 


0-216 
0-183 
0-150 


20 
21 
21 


58 
2 
6 


11-37 
7 92 
4-48 


Tues. 
Wed. 
Thur. 


7 
8 
9 


21 
21 

21 


24 

28 
32 


23-63 
22-60 
20-79 


9-974 
9-941 
9-909 


15 
14 
14 


14 
55 
36 


25-4 
29-1 

17-7 


+47 '03 
47-66 

48-28 


14 22-60 
14 25-01 
14 26-65 


0-117 
0-084 
0-052 


21 

21 
21 


10 
13 

17 


1-03 
57-59 
54-14 


Frid. 
Sat. 

•Sun. 


10 
11 
12 


21 
21 
21 


36 
40 
44 


18-21 

14-88 
10-80 


9-877 
9-846 
9-815 


14 

13 
13 


16 

57 
37 


51-6 
11 2 
16-9 


48-88 
49-47 
50-03 


14 27-51 
14 27-63 
14 26-99 


0-020 
0-011 
0-042 


21 
21 
21 


21 
25 
29 


50-70 
47-25 
43-81 


Mon. 
'Tues. 
Wed. 


13 
14 

15 


21 
21 

21 


48 
52 
55 


5-98 

0-43 

54-16 


9-784 
9-753 
9-723 


13 
12 
12 


17 
56 

36 


9-1 

48-3 
14-9 


4-50-59 
51-12 
51-65 


14 25-63 
14 23-52 
14 20-70 


0-073 
0-104 
0-134 


21 
21 
21 


33 

37 

41 


40-35 
36-91 
33 -4b 


Thur. 
Frid. 
.Sat. 


16 
17 
18 


21 
22 

22 


59 
3 

7 


47-17 
39-47 
31-07 


9-693 
9-664 
9-635 


12 
11 
11 


15 
51 
33 


29-3 
32-1 
23-6 


+52-14 
52-62 
53-07 


14 17-15 
14 12-90 
14 7-94 


0-164 
0-193 
0-222 


21 
21 
21 


45 
49 

53 


30-02 
26-57 
23-13 



Of the same general nature with the Ephemeris are catalogues of 
the fixed stars. The object of such a catalogue is to give the right 
ascension and declination of a number of stars for some epoch, the 
beginning of the year 1875 for instance, with the data by which the 
position of a star can be found at any other epoch. " Such cata- 
logues are, however, imperfect owing to tlie constant small changes 
in the positions of the stars and the errors and imperfections of the 
older observations. In consequence of these imperfections, a consid- 
erable part of the work of the astronomer engaged in accurate de- 
terminations of geographical positions consist in rinding the most 
.accurate positions of the stars which he makes use of. 



part n. 

THE SOLAR SYSTEM IS DETAIL 



CHAPTER I. 

STRUCTURE OF THE SOLAR SYSTEM. 

The solar system, as it is known to us through the dis- 
coveries of Copernicus, Kepler, Newton and their suc- 
cessors, consists of the sun as a central hody, around which 
revolve the major and minor planets, with their satellites, 
a few periodic comets, and an unknown number of meteor 
swarms. These are permanent members of the system. 
At times other comets appear, and move usually in par- 
abolas through the system, around the sun, and away from 
it into space again, thus visiting the system without be- 
ing permanent members of it. 

The bodies of the system may be classified as follows : 

1. The central body — the Sun. 

2. The four inner planets — Mercury ', Venus, the J?arth, 
Mars. 

3. A group of small planets, sometimes called Asteroids, 
revolving outside of the orbit of Mars. 

4. A group of four outer planets — Jupiter, Saturn, 
Uranus, and JVeptune. 

5. The satellites, or secondary bodies, revolving about 
the planets, their primaries. 

6. A number of comets and meteor swarms revolving 
in very eccentric orbits about the Sun. 



268 ASTRONOMY. 

The eight planets of Groups 2 and 4 are sometimes 
classed together as the major planets, to distinguish them 
from the two hundred or more minor planets of Group 3. 
The formal definitions of the various classes, laid down 
by Sir "William Herschel in 1802, are worthy of repe- 
tition : 

Planets are celestial bodies of a certain very consider- 
able size. 

They move in not very eccentric ellipses about the 
sun. 

The planes of their orbits do not deviate many degrees 
from the plane of the earth's orbit. 

Their motion about the sun is direct. 

They may have satellites or rings. 

They have atmospheres of considerable extent, which, 
however, bear hardly any sensible proportion to their 
diameters. 

Their orbits are at certain considerable distances from 
each other. 

Asteroids, now more generally known as small or 
minor planets, are celestial bodies which move about the 
sun in orbits, either of little or of considerable eccen- 
tricity, the planes of which orbits may be inclined to the 
ecliptic in any angle whatsoever. They may or may not 
have considerable atmospheres. 

Comets are celestial bodies, generally of a very small 
mass, though how far this may be limited is yet un- 
known. 

They move in very eccentric ellipses or in parabolic 
arcs about the sun. 

The planes of their motion admit of the greatest variety 
in their situation. 

The direction of their motion is also totally undeter- 
mined. 

They have atmospheres of very great extent, which 
show themselves in various forms as tails, coma, haziness, 
etc. 



MAGNITUDES OF THE PLANETS. 



269 



Belative Sizes of the Planets.— The comparative sizes of 
the major planets, as they would appear to an observer 
situated at an equal distance from all of them, is given in 
the following figure. 




Fig. 74. — relative sizes of the planets. 

The relative apparent magnitudes of the sun, as seen 
from the various planets, is shown in the next figure. 

Flora and Mnemosyne are two of the asteroids. 

A curious relation between the distances of the planets, 
known as Bode's law, deserves mention. If to the num- 
bers, 

0, 3, 6, 12, 24, 48, 90, 192, 3S4, 



270 ASTRONOMY. 

each of which (the second excepted) is twice the preced- 
ing, we add 4, we obtain the series, 

4, 7, 10, 16, 28, 52, 100, 196, 388. 

These last numbers represent approximately the dis- 




Fig. 75. — apparent magnitudes of the sun as seen from dip- 
ferent planets. 

tances of the planets from the sun (except for Neptune, 
which was not discovered when the so-called law was an- 
nounced). 

This is shown in the following table : 



CHARACTERISTICS OF THE PLANETS. 



271 



Planets. 


Actual 
Distance. 


Bode's Law. 


Mercury 


39 

7-2 

10-0 

15-2 

27-7 

52-0 

95-4 

191-8 

300-4 


4-0 


Venus 


7-0 


Earth 


10-0 


Mars 


16-0 


[Ceres] 


28-0 


Jupiter 


52-0 


Saturn 


100.0 


Uranus 


196-0 


Neptune 


388-0 







It will be observed that Neptune does not fall within 
this ingenious scheme. Ceres is one of the minor planets. 

The relative brightness of the sun and the various 
planets has been measured by Zollner, and the results 
are given below. The column per cent shows the per- 
centage of error indicated in the separate results : 



Sun and 


Ratio : 1 to 


Percent, of Error. 


Moon 


618,000 

6,994,000,000 

5,472,000,000 

130,980,000,000 

8,486,000,000,000 

79,620,000,000,000 


1-6 


Mars 


5-8 


Jupiter 


5-7 


Saturn (ball alone) 

Uranus 


5-0 
6-0 


Neptune 


5*5 







The differences in the density, size, mass and distance 
of the several planets, and in the amount of solar light 
and heat which they receive, are immense. The distance 
of JVeptime is eighty times that of Mercury, and it re- 
ceives only -g-^ou as mucn light and heat from the sun. 
The density of the earth is about six times that of water, 
while Saturn's mean density is less than that of water. 

The mass of the sun is far greater than that of any 
single planet in the system, or indeed than the combined 
mass of all of them. In general, it is a remarkable fact 
that the mass of any given planet exceeds the sum of the 
masses of all the planets of less mass than itself. This is 



272 



ASTRONOMY. 



shown in the following table, where the masses of the plan- 
ets are taken as fractions of the sun's mass, which we here 
express as 1,000,000,000: 



a 
8 

9 
3 


B 

at 


c 




09 


C 

s 


B 
S 
OS 


u 

V 

ft 


02 


Planets. 


200 

. 


324 


2,353 


3,060 


44,250 


51,600 


285,580, 


954,305 


1,000,000,000 


Masses. 



200 < 



324 



524 < 


2,353 


2,877 < 


3,060 


5,937 < 


44,250 


50,187 < 


51,600 


101,787 < 


285,580 


387,367 < 


954,305 


341,672 < 


1,000,000,000 



The mass of Mercury is less than the mass ) 
of Mars : \ 

The sum of masses of Mercury and Mars ) 
is less than the mass of Venus : j" 

Mercury + Mars + Venus < Earth : 

Mercury + Mars + Venus + Earth < Ura- ) 
nus : J 

Mercury + Mars -+- Venus + Earth + Ura- ) 
nus < Neptune : j 

Mercury + Mars + Venus + Earth + Ura- 
nus + Neptune < Saturn : 

Mercury + Mars 4- Venus -+- Earth -+- Ura- 
nus + Neptune + Saturn < Jupiter : 

Combined mass of all the planets is less ) 1 
than that of the Sun : \ 

The total mass of the small planets, like their number, 
is unknown, but it is probably less than one thousandth 
that of our earth, and would hardly increase the sum total 
of the above masses of the solar system by more than one 
or two units. The sun's mass is thus over 700 times that 
of all the other bodies, and hence the fact of its central 
position in the solar system is explained. In fact, the 
centre of gravity of the whole solar system is very little 
outside the body of the sun, and will be inside of it when 
Jupiter and Saturn are in opposite directions from it. 

Planetary Aspects. — The motions of the planets about 
the sun have been explained in Chapter IV. From what 
Is there said it appears that the best time to see one of the 



PLANETARY ASPECTS. 



273 



outer planets will be when it is in opposition — that is, when 
its geocentric longitude or its right ascension differs 180° 
or 12 h from that of the sun. At such a time the planet 
will rise at sunset and culminate at midnight. During the 
three months following opposition, the planet will rise from 
three to six minutes earlier every day, so that, knowing 
when a planet is in opposition, it is easy to find it at any 
other time. For example, a month after opposition the 




planet will be two to three hours high about sunset, and 
will culminate about nine or ten o'clock. Of course the 
inner planets never come into opposition, and hence are 
best seen about the times of their greatest elongations. 

The above figure gives a rough plan of part of the 
solar system as it would appear to a spectator immediately 
above or below the plane of the ecliptic. 



274 ASTRONOMY. 

It is drawn approximately to scale, the mean distance of 
the earth (= 1) being half an inch. The mean distance of 
Saturn would be 4*77 inches, of 'Uranus 9*59 inches, of 
Xeptune 15 -03 inches. On the same scale the distance of 
the nearest fixed star would be 103,133 inches, or over one 
and one half miles. 

The arrangement of the planets and satellites is then — 

The Inner Group. Asteroids. The Outer Group. 

Mercurv. ) minnir ^ OT1Qt -. l' Jupiter and 4 moons. 

■r- * / 200 minor planets, \ e 5 •, 

Venus. ( , ___£. w _ ' Saturn and 8 moons. 

Earth and Moon. f manv more 1 Uranus and 4 moons. 

Mars and 2 moons. ) ' ore. ^ Neptune and 1 moon. 

To avoid repetitions, the elements of the major planets 
and other data are collected into the two following tables, 
to which reference may be made by the student. The 
nnits in terms of which the various quantities are given 
are those familiar to us, as miles, days, etc., yet some of 
the distances, etc., are so immensely greater than any 
known to our daily experience that we must have recourse 
to illustrations to obtain any idea of them at all. For ex- 
ample, the distance of the sun is said to be 92 h million 
miles. It is of importance that some idea should be had 
of this distance, as it is the unit, in terms of which not 
only the distances in the solar system are expressed, but 
which serves as a basis for measures in the stellar universe. 
Thus when we say that the distance of the stars is over 
200,000 times the mean distance of the sun, it becomes 
necessary to see if some conception can be obtained of one 
factor in this. Of the abstract number, 92,500,000, we 
have no conception. It is far too great for us to have 
counted. We have never taken in at one view, even 
a million similar discrete objects. To count from 1 to 
200 requires, with very rapid counting, 60 seconds. Sup- 
pose this kept up for a day without intermission ; at the 
end we should have counted 283,000, which is about -g-J^ 
of 92,500,000. Hence over 10 months' uninterrupted 
counting by night and day would be required simply to 
enumerate the miles, and long before the expiration of 






EXTENT OF THE SOLAR SYSTEM. 275 

the task all idea of it would have vanished. We may take 
other and perhaps more striking examples. We know, 
for instance, that the time of the fastest express-trains be- 
tween New York and Chicago, which average 40 miles per 
hour, is about a day. Suppose such a train to start for 
the sun and to continue running at this rapid rate. It 
would take 363 years for the journey. Three hundred 
and sixty- three years ago there was not a European settle- 
ment in America. 

A cannon-ball moving continuously across the interven- 
ing space at its highest speed would require about nine 
jears to reach the sun. The report of the cannon, if it 
could be conveyed to the sun with the velocity of sound in 
&ir, would arrive there five years after the projectile. 
Such a distance is entirely inconceivable, and yet it is 
only a small fraction of those with which astronomy has to 
■deal, even in our own system. The distance of Neptune 
is 30 times as great. 

If we examine the dimensions of the various orbs, we meet 
-almost equally inconceivable numbers. The diameter 
of the sun is 860,000 miles ; its radius is but 430,000, and 
yet this is nearly twice the mean distance of the moon 
from the earth. Try to conceive, in looking at the moon 
in a clear sky, that if the centre of the sun could be 
placed at the centre of the earth, the moon would be far 
•within the sun's surface. Or again, conceive of the force 
of gravity at the surface of the various bodies of the sys- 
tem. At the sun it is nearly 28 times that known to us. 
A pendulum beating seconds here would, if transported 
to the sun, vibrate with a motion more rapid than that of 
a watch-balance. The muscles of the strongest man would 
not support him erect on the surface of the sun : even 
lying down he would crush himself to death under his 
own weight of two tons. We may by these illustrations 
get some rough idea of the meaning of the numbers in 
these tables, and of the incapability of our limited ideas to 
comprehend the true dimensions of even the solar system. 



276 



ASTRONOMY. 





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CHAPTER II. 

THE SUN". 
§ 1. GENERAL SUMMARY. 

To the student of the present time, armed with the 
powerful means of research devised by modern science, 
the sun presents phenomena of a very varied and complex 
character. To enable the nature of these phenomena to be 
clearly understood, we preface our account of the physical 
constitution of the sun by a brief summary of the main 
features seen in connection with that body. 

Photosphere. — To the simple vision the sun presents 
the aspect of a brilliant sphere. The visible shining sur- 
face of this sphere is called the photosphere, to distinguish 
it from the body of the sun as a whole. The apparently 
flat surface presented by a view of the photosphere is called 
the sun's dish. 

Spots. — When the photosphere is examined with a tele- 
scope, small dark patches of varied and irregular outline 
are frequently found upon it. These are called the solar 
spots. 

Rotation. — "When the spots are observed from day to 
day, they are found to move over the sun's disk in such a 
way as to show that the sun rotates on its axis in a period 
of 25 or 26 days. The sun, therefore, has axis, poles, and 
equator, like the earth, the axis being the line around 
which it rotates. 

Faculae. — Groups of minute specks brighter than the 
general surface of the sun are often seen in the neighbor- 
hood of spots or elsewhere. They are called faculw: 



FEATURES OF THE SUN. 279 

Chromosphere, or Sierra. — The solar photosphere is 
covered by a layer of glowing vapors and gases of very ir- 
regular depth. At the bottom lie the vapors of many 
metals, iron, etc., volatilized by the fervent heat which 
reigns there, while the upper poitions are composed prin- 
cipally of hydrogen gas. This vaporous atmosphere is 
commonly called the chromosphere, sometimes the sierra. 
It is entirely invisible to direct vision, whether with the 
telescope or naked eye, except for a few seconds about 
the beginning or end of a total eclipse, but it may be seen 
on any clear day through the spectroscope. 

Prominences, Protuberances, or Red Plames. — The 
gases of the chromosphere are frequently thrown up in 
irregular masses to vast heights above the photosphere, it 
may be 50,000, 100,000, or even 200,000 kilometres. 
Like the chromosphere, these masses have to be studied 
with the spectroscope, and can never be directly seen ex- 
cept w T hen the sunlight is cut off by the intervention of the 
moon during a total eclipse. They are then seen as rose- 
colored flames, or piles of bright red clouds of irregular 
and fantastic shapes. They are now usually called ' ' prom- 
inences" by the English, and "protuberances" by 
French writers. 

Corona. — During total eclipses the sun is seen to be en- 
veloped by a mass of soft wdiite light, much fainter than 
the chromosphere, and extending out on all sides far be- 
yond the highest prominences. It is brightest around the 
edge of the sun, and fades off toward its outer boundary, 
by insensible gradations. This halo of light is called the 
corona, and is a very striking object during- a total eclipse. 

§ 2. THE PHOTOSPHERE. 

Aspect and Structure of the Photosphere. — The disk 
of the sun is circular in shape, no matter what side of the 
sun's globe is turned toward us, whence it follows that the 
sun itself is a sphere. The aspect of the disk, when 



280 ASTRONOMY. 

viewed with tlie naked eye, or with a telescope of 
low power, is that of a uniform bright, shining surface, 
hence called the photosphere. With a telescope of 
higher power the photosphere is seen to be diversified 
with groups of spots, and under good conditions the 
whole mass has a mottled or curdled appearance. This 
mottling is caused by the presence of cloud-like forms, 
whose outlines though faint are yet distinguishable. 
The background is also covered with small white dots 
or forms still smaller than the clouds. These are the 
" rice-grains, " so called. The clouds themselves are 
composed of small, intensely bright bodies, irregularly 
distributed, of tolerably definite shapes, which seem to be 
suspended in or superposed on a darker medium or back- 
ground. The spaces between the bright clots vary in 
diameter from 2" to 4" (about 1400 to 2800 kilome- 
tres). The rice-grains themselves have been seen to 
be composed of smaller granules, sometimes not more 
than 0"-3 (135 miles) in diameter, clustered together. 
Thus there have been seen at least three orders of 
aggregation in the brighter parts of the photosphere : 
the larger cloud -like forms ; the rice grains ; and, small- 
est of all, the granules. These forms have been studied 
with the telescope by Secchi, Huggins, and Langley, 
and their relations tolerably well made out. 

In the Annuaire of the Bureau of Longitudes for 1878 (p. C89), 
M. Janssen gives an account of his recent discovery of the reticulated 
arrangement of the solar photosphere. The paper is accompanied 
by a photograph of the appearances described, which is enlarged 
threefold. Photographs less than four inches in diameter cannot 
satisfactorily show such details. As the granulations of the solar 
surface are, in general, not greatly larger than \" or 2", the photo- 
graphic irradiation, which is sometimes 20" or more, may completely 
obscure their characteristics. This difficulty M. Janssen has over- 
come by enlarging the image and shortening the time of expos- 
ure. In this way the irradiation is diminished, because as the di- 
ameters increase, the linear dimensions of the details are increased, 
and " the imperfections of the sensitive plate have less relative im- 
portance." 



THE SUN'S PHOTOSPHERE. 281 

Again, M. Janssen has noted that in short exposures the photo- 
graphic spectrum is almost monochromatic. 

In this way it differs greatly from the visible spectrum, and to 
the advantage of the former for this special purpose. The diameter 
of the solar photograms have since 1874 been successively increased 
to 12, 15, 20, and 30 centimetres. The exposure is made equal all 
over the surface. In summer this exposure for the largest photo- 




FlG. 77. — RETICULATED ARRANGEMENT OF THE PHOTOSPHERE. 

grams is less than 9, 0005. The development of such pictures is 
very slow. 

These photograms, on examination, show that the solar surface is 
covered with a fine granulation. The forms and the dimensions of 
the elementary surfaces are very various. They vary in size from 
0"-3 or 0"-4 to 3'' or 4" (200 to 3000 kilometres).' Their forms 



282 ASTRONOMY. 

are generally circles or ellipses, but these curves are sometimes 
greatly altered. This granulation is apparently spread equally all 
over the disk. The brilliancy of the points is very variable, and 
they appear to be situated at different depths below the photo- 
sphere : the most luminous particles, those to which the solar light 
is chiefly due, occupy only a small fraction of the solar surface. 

The most remarkable feature, however, is ' ' the reticulated ar- 
rangement of the parts of the photosphere." "The photograms 
show that the constitution of the photosphere is not uniform 
throughout, but that it is divided in a series of regions more or 
less distant from each other, and having each a special constitution. 
These regions have, in general, rounded contours, but these are 
often almost rectilinear, thus forming polygons. The dimensions 
of these figures are very variable ; some are even V in diameter 
(over 25,000 miles)." "Between these figures the graiDS are 
sharply defined, but in their interior they ai'3 almost effaced and 
run together as if by some force. ' ' These phenomena can be best 
understood by a reference to the figure of M. Ja^ssen (p. 281). 

Light and Heat from the Photosphere. — The photo- 
sphere is not equally bright all over the apparent disk. 
This is at once evident to the eye in observing the sun with 
a telescope. The centre of the disk is most brilliant, and 
the edges or limbs are shaded off so as to forcibly suggest 
the idea of an absorptive atmosphere, which, in fact, is the 
cause of this appearance. 

Such absorption occurs not only for the rays by winch 
we see the sun, the so-called visual rays, but for those 
which have the most powerful effect in decomposing the 
salts of silver, the so-called chemical rays, by which the 
ordinary photograph is taken. 

The amount of heat received from different portions of 
the sun's disk is also variable, according to the part of 
the apparent disk examined. This is what we should ex- 
pect. That is, if the intensity of any one of these radiations 
(as felt at the earth) varies from centre to circumference, 
that of every other should also vary, since they are all 
modifications of the same primitive motion of the sun's 
constituent particles. But the constitution of the sun's 
atmosphere is such that the law of variation for the three 
classes is different. The intensity of the radiation in the 
sun itself and inside of the absorptive atmosphere is prob- 



SOLAR RADIATION. 



283 



ably nearly constant. The ray which leaves the centre of 
the sun's disk in passing to the earth, passes through the 
smallest possible thickness of the solar atmosphere, while 
the rays from points of the sun's body which appear to 
us near the limbs pass, on the contrary, through the maxi- 
mum thickness of atmosphere, and are thus longest sub- 
jected to its absorptive action. 

This is plainly a rational explanation, since the part of 
the sun which is seen by us as the limb varies with the 
position of the earth in its orbit and with the position of 
the sun's surface in its rotation, and has itself no physical 
peculiarity. The various absorptions of different classes 
of rays correspond to this supposition, the more refrangi- 
ble rays suffering most absorption, as they must do, being 
composed of waves of shorter wave ]ength. 

The following table gives the observed ratios of the amount of 
heat, light, and chemical action at the centre of the sun and at 
various distances from the centre toward the limb. The first 
column of the table gives the apparent distances from the centre 
of the disk, the sun's radius being 1*00. The second column gives 
the percentage of heat-rays received by an observer on the earth 
from points at these various distances. That is, for every 100 heat- 
rays reaching the earth from the sun's centre, 95 reach us from a 
point half way from the centre to the limb, and so on. 

Analogous data are given for the light-rays and the chemical 
rays. The data in regard to heat are due to Professor Langley ; 
those in regard to light and chemical action to Professor Pickering 
and Pr. Vogel respectively. 



Distance from 
Centre. 


Heat Rays. 


Light Rays. 


Chemical Rays. 


0-00 


100 
99 
95 

86 

*62 
50 


100 

97 
91 
79 
69 
55 

'37' 


100 
98 
90 

4S 
25 
23 
18 
13 


0-25 


0-50 


0-75 


0-85 


0-95 


0-96 


0-98 


1-00 





For two equal apparent surfaces, A near the sun's centre and J5 
near the limb, we may say that the rays from the two surfaces when 



284 ASTRONOMY. 

received at the earth have approximately the following relative 
effects : 

A has twice as much effect on a thermometer as B (heat); 

A has three times as much illuminating effect as B (light); 

A has seven times as much effect in decomposing the photo- 
graphic salts of silver as B (actinic effect). 

It is to be carefully borne in mind that the above numbers refer 
to variations of the sun's rays received from different equal surfaces 
A and B, in their effect upon certain arbitrary terrestrial standards of 
measure. If, for example, the decomposition of other salts than 
those employed for ordinary photographic work be taken as stand- 
ards, then the numbers will be altered, and so on. We are simply 
measuring the power of solar rays selected from different parts of 
the sun's apparent disk, and hence exposed to different conditions 
of absorption in his atmosphere, to do work of a certain selected 
kind, as to raise the temperature of a thermometer, to affect the 
human retina, or to decompose certain salts of silver. 

In this the absorption of the earth's atmosphere is rendered con- 
stant for each kind of experiment. This atmosphere has, however, 
a very strong absorptive effect. We know that we can look at the 
setting or rising sun, which sends its light rays through great 
depths of the earth's atmosphere, but not upon the sun at noon- 
day. The temperature is lower at sunrise or at sunset than at noon, 
and the absorption of chemical rays is so marked that a photograph 
of the solar spectrum which can be taken in three seconds at noon 
requires six hundred seconds about sunset — that is, two hundred 
times as long (Draper). 

Amount of Heat Emitted by the Sun. — Owing to the 
absorption of the solar atmosphere, it follows that we re- 
ceive only a portion — perhaps a very small portion — of 
the rays emitted by the sun's photosphere. 

If the sun had no absorptive atmosphere, it would seem 
to us hotter, brighter, and more blue in color. 

Exact notions as to how great this absorption is are hard 
to gain, but it may be said roughly that the best authori- 
ties agree that although it is quite possible that the sun's 
atmosphere absorbs half the emitted rays, it probably does 
not absorb four fifths of them. 

It is a curious, and as yet we believe unexplained fact, 
that the absorption of the solar atmosphere does not affect 
the darkness of the Fraunhof er lines. They seem equally 
black at the centre and edge of the sun.* The amount 

* Prof. Young has spoken of a slight observable difference. 



HEAT OF THE SUN. 285 

of this absorption is a practical question to us on the earth. 
So long as the central body of the sun continues to emit 
the same quantity of rays, it is plain that the thickness of 
the solar atmosphere determines the number of such rays 
reaching the earth. If in former times this atmosphere 
was much thicker, then less heat would have reached the 
earth. Professor Langley suggests that the glacial epoch 
may be explained in this way. If the central body of the 
sun has likewise had different emissive powers at different 
times, this again would produce a variation in the tempera- 
ture of the earth. 

Amount of Heat Radiated. — There is at present no way 
of determining accurately either the absolute amount of 
heat emitted from the central body or the amount of this 
heat stopped by the solar atmosphere itself. All that can 
be done is to measure (and that only roughly) the amount 
of heat really received by the earth, without attempting to 
define accurately the circumstances which this radiation 
has undergone before reaching the earth. 

The difficulties in the way of determining how much 
heat reaches the earth in any definite time, as a year, are 
twofold. First, we must be able to distinguish between 
the heat as received by a thermometric apparatus from 
the sun itself and that from external objects, as our own 
atmosphere, adjacent buildings, etc.; and, second, we 
must be able to allow for the absorption of the earth's 
atmosphere. 

Pouillet has experimented upon this question, making 
allowance for the time that the sun is below the horizon 
of any place, and for the fact that the solar rays do not in 
general strike perpendicularly but obliquely upon any 
given part of the earth's surface. His conclusions may 
be stated as follows : if our own atmosphere were re- 
moved, the solar rays would have energy enough to melt 
a layer of ice 9 centimetres thick over the whole earth 
daily, or a layer of about 32 metres thick in a year. 

Of the total amount of heat radiated by the sun, tho 



286 ASTRONOMY. 

earth receives but an insignificant share. The sun is 
capable of heating the entire surface of a sphere "whose ra- 
dius is the earth's mean distance to the same degree that 
the earth is now heated. The surface of such a sphere is 
2,170,000,000 times greater than the angular dimensions 
of the earth as seen from the sun, and hence the earth re- 
ceives less than one two billionth part of the solar radia- 
tion. The rest of the solar rays are, it is generally believed, 
lost in space, so far as the human race is concerned. 

It is found, from direct measures, that a sun-spot gives less heat, 
area for area, than the unspotted photosphere, and it is an interest- 
ing question how much the climate of the earth can be affected by 
this difference. 

Professor Langley, of Pittsburgh, has made measurements of the 
direct effect of sun-spots on terrestrial temperature. The observa- 
tions consisted in measuring the relative amounts of umbral, penum- 
bral, and photospheric radiation. The relative umbral, penumbral, 
and photospheric areas were deduced from the Kew observations of 
spots ; and from a consideration of these data, and confining the 
question strictly to changes of terrestrial temperature due to this 
cause alone, Langley deduces the result that " sun-spots do ex- 
ercise a direct effect on terrestrial temperature by decreasing the 
mean temperature of the earth at their maximum." This change 
is, however, very small, as "it is represented by a change in the 
mean temperature of our globe in eleven years not greater than 
0*5° C, and not less than - 3° C." It is not intended to show that 
the earth is, on the whole, cooler in maximum sun-spot years, but 
that, as far as this cause goes, it tends to make the earth cooler by 
this minute amount. What other causes may co-exist with the 
maximum spot-frequency are not considered. 

Solar Temperature. — From the amount of heat actually 
radiated by the sun, attempts have been made to determine 
the actual temperature of the solar surface. The esti- 
mates reached by various authorities differ widely, as the 
laws which govern the absorption within the solar en- 
velope are almost unknown. Some such law of absorp- 
tion has to be supposed in any such investigation, and the 
estimates have differed widely according to the adapted 
law. i 

Secchi estimates this temperature as about 6,100,000° C. 
Other estimates are far lower, but, according to all sound 



SPOTS ON THE SUN. 287 

philosophy, the temperature must far exceed any ter- 
restrial temperature. There can be no doubt that if the 
temperature of the earth's surface were suddenly raised to 
that of the sun, no single chemical element would remain 
in its present condition. The most refractory materials 
would be at once volatilized. 

We may concentrate the heat received upon several square feet 
(the surface of a huge burning-lens or mirror, for instance), 
examine its effects at the focus, and, making allowance for the con- 
densation by the iens, see what is the minimum possible tempora- 
ture of the sun. The temperature at the focus of the lens cannot 
be higher than that of the source of heat in the sun ; we can only 
concentrate the heat received on the surface of the lens to one 
point and examine its effects. If a lens three feet in diameter be 
used, the most refractory materials, as fire-clay, platinum, the dia- 
mond, are at once melted or volatilized. The effect of the lens is 
plainly the same as if the earth were brought closer to the sun, in 
the ratio of the diameter of the focal image to that of the lens. In 
the case of the lens of three feet, allowing for the absorption, etc., 
this distance is yet greater than that of the moon from the earth, 
so that it appears that any comet or planet so close as this to the 
sun, if composed of materials similar to those in the earth, must 
be vaporized. 

If we calculate at what rate the temperature of the sun would be 
lowered annually by the radiation from its surface, we shall find it 
to be 134* Centigrade yearly if its specific heat is that of water, 
and between 3° and 6° per annum if its specific heat is the same as 
that of the various constituents of the earth itself. It would there- 
fore cool down in a few thousand years by an appreciable amount. 

§ 3. SUN-SPOTS AND PACUL^J. 

A very cursory examination of the sun's disk with a 
small telescope will generally show one or more dark spots 
upon the photosphere. These are of various sizes, from 
minute black dots 1" or 2" in diameter (1000 kilometres 
or less) to large spots several minutes of arc in extent. 

Solar spots generally have a dark central nucleus or 
umbra, surrounded by a border or penumbra of grayish 
tint, intermediate in shade between the central blackness 
and the bright photosphere. By increasing the power of 
the telescope, the spots are seen to be of very complex 
forms. The umbra is often extremely irregular iti shape, 



288 



ASTRONOMY. 



and is sometimes crossed by bridges or ligaments of shining 
matter. The jpenumbra is composed of filaments of 
brighter and darker light, which are arranged in striae. 
The appearances of the separate filaments are as if they 
were directed downward toward the interior of the spot 
in an oblique direction. The general aspect of a spot un- 
der considerable magnifying power is shown in Fig. 78. 

The first printed account of solar spots was given by 
Fabkitius in 1611, and Galileo in the same year (May, 
1611) also described them. They were also attentively 




UMBRA AND PENUMBRA OF SUN-SPOT. 



studied by the Jesuit Scheiner, who supposed them to be 
small planets projected against the solar disk. This idea 
was disproved by Galileo, whose observations showed 
them to belong to the sun itself, a ad to move uniformly 
across the solar disk from east to west. A spot just visible 
at the east limb of the sun omany one day travelled slowly 
across the disk for 12 or 14 days, when it reached the west 
limb, behind which it disappeared. After about the same 
period, it reappeared at the eastern limb, unless, as is often 
the case, it had in the mean time vanished. 



SUITS SPOTS AND ROTATION. 289 

The spots are not permanent in their nature, but are 
formed somewhere on the sun, and disappear after lasting 
a few days, weeks, or months. But so long as they last 
they move regularly from east to west on the sun's appar- 
ent disk, making one complete rotation in about 25 days. 
This period of 25 days is therefore approximately the rota- 
tion period of the sun itself. 

Spotted Region. — It is found that the spots are chiefly con- 
fined to two zones, one in each hemisphere, extending from about 
10° to 35° or 40° of heliographic latitude. In the polar regions, 
spots are scarcely ever seen, and on the solar equator they are much 




Fig. 79.— photograph op the sun. 

more rare than in latitudes 10° north or south. Connected -with 
the spots, but lying on or above the solar surface, are faculce, mot- 
tlings of light brighter than the general surface of the sun. The 
formation of a sun-spot is said to be often presaged by the ap- 
pearance of faculse near the point where the spot is to form. 

Solar Rotation. — To obtain the exact period of rotation, the 
spots must be carefully fixed in position by micrometric measures 
from day to day, the times of the measures being noted. Better 
still, daily photographs may be made and afterward measured. 
This has been done by several observers, and the remarkable result 
reached that the spots do not all rotate exactly in the same period, 
but that this time, as determined from any spot, depends upon the 
JieliograpMc latitude of the spot, or its angular distance from the 



290 ASTRONOMY. 

solar equator. A series of observations made by Mr. Carrington' 
of England (by the eye) give the following values of the rotation 
times T, for spots in different heliographic latitudes L : 



L= 0° 


5° 


10° 


15° 


20° 


T= 25-187 days 


25-222 


25-327 


25-500 


25-730 


L= 25° 


30° 


35° 


40° 


45° 


T= 26-040 


26-398 


26-804 


27-252 


27-730 



The period of rotation seems also to vary somewhat in different 
years even for spots in the same heliographic latitude, so that we 
really cannot assign any one definite rotation time to the sun, as 
we can to the earth or the moon. 

"The probability is that the sun, not being solid, has really no one 
period of rotation, but different portions of its surface and of its in- 
ternal mass move at different rates, and to some extent independent- 
ly of each other, though approximately in one plane inclined abont 
7° to the ecliptic, and around a common axis. The individual 
spots drift in latitude as well as in longitude, and, on the whole, it 
appears that spots within 15° or 20° of the solar equator on either 
side move toward the equator, while beyond this limit they move 
away friom it." (Young.) 

Solar Axis and Equator. — The spots must revolve with the 
surface of the sun about his axis, and the directions of their motions 
must be approximately parallel to his equator. Fig. 80 shows 
tlie appearances as actually observed, the dotted lines representing 
the apparent paths of the spots across the sun's disk at different 
times oif the year. In June and December these paths, to an ob- 
server on the earth, seem to be right lines, and hence at these times 
the observer must be in the plane of the solar equator. At other 
times the paths are ellipses, and in March and September the 
planes of these ellipses are most oblique, showing the spectator to 
be then furthest from the plane of the solar equator. The incli- 
nation of the solar equator to the ecliptic is, as already stated, about 
7° 9', and the axis of rotation is of course perpendicular to it. 

Nature of the Spots. — The sun-spots are really depres- 
sions in the jxhotospkere, as was first pointed out by An- 
drew Wilson of Glasgow. When a spot is seen at the 
edge of the disk, it appears as a notch in the limb, and is 
elliptical in shape. As the rotation carries it further and 
further on to the disk, it becomes more and more nearly 
circular in shape, and after passing the centre of the disk 
the appearances take place in reverse order. 

These observations were explained by Wilson, and more fully by 
Sir William Herschel, by supposing the sun to consist of an in- 
terior dark cool mass, surrounded by two layers of clouds. The 



SOLAR SPOTS. 291 

outer layer, which forms the visible photosphere, was supposed 
extremely brilliant. The inner layer, which could not be seen 
except when a cavity existed in the photosphere, was supposed 
to be dark. The appearance of the edges of a spot, which has 
been described as the penumbra, was supposed to arise from 
those dark clouds. The spots themselves are, according to this 
view, nothing but openings through both of the atmospheres, the 



Fig. 80. — apparent path of solak spot at different seasons. 

nucleus of the spot being simply the black surface of the inner 
sphere of the sun itself. 

This theory, which the figure on the next page exemplifies, 
accounts for the facts as they were known to Herschel. But when 
it is confronted with the questions of the cause of the sun's heat 
and of the method by which this heat has been maintained con- 
stant in amount for centuries, it breaks down completely. The 



292 



ASTRONOMY. 



conclusions of "Wilson and Herschel, that the spots are depressions 
in the sun's surface, are undoubted. But the existence of a cool cen- 
tral and solid nucleus to the sun is now known to be impossible. 
The apparently black centres of the spots are so mostly by contrast. 
If they were seen against a perfectly black background, they would 
appear very bright, as has been proved by the photometric measures 
of Professor Langley. And a cool solid nucleus beneath such an 
atmosphere as Herschel supposed would soon become gaseous by 
the conduction and radiation of the heat of the photosphere. The 
supply of solar heat, which has been very nearly constant during 
the historic period, would in a sun so constituted have sensibly 
diminished in a few hundred years. For these and other reasons, 
the hypothesis of Herschel must be modified, save as to the fact 
that the spots are really cavities in the photosphere. 




Fig. 81. — appearance of a spot near the lolb and near this 
centre op the sun. 

Number and Periodicity of Solar Spots. — The number 
of solar spots which come into view varies from year to 
year. Although at first sight this might seem to be what 
we call a purely accidental circumstance, like the occur- 
rence of cloudy and clear years on the earth, yet the series 
of observations of sun-spots by Hofrath Schwabe of 
Dessau (see the table), continued by him for forty years, 
established the fact that this number varied periodically. 
This had indeed been previously suspected by Horrebow, 



PERIODICITY OF SUN-SPOTS. 



293 



but it was independently suggested and completely proved 

by Schwabe. 

Table of Schwabe's Results. 



Year. 


Days of 
Observation. 


Days of no 
Spots. 


New Groups. 


Mean Diurnal 
Variation in 

Declination of 

the Magnetic 

Needle. 


1826 


277 
273 
282 
244 
217 
239 
270 
247 
273 
244 
200 
168 
202 
205 
263 
283 
307 
312 
321 
332 
314 
276 
278 
285 
308 
308 
337 
299 
334 
313 
321 
324 
335 
343 
332 
322 
317 
330 
325 
307 
349 
316 
301 


22 
2 


1 
3 

49 
139 
120 

18 




3 

15 

64 
149 
111 

29 
1 



2 

2 
3 

65 
146 
193 

52 




3 
2 
4 

25 

76 
195 

23 


118 

161 

225 

199 

190 

149 

84 

33 

51 

173 

272 

333 

282 

162 

152 

102 

68 

34 

52 

114 

157 

257 

330 

238 

186 

151 

125 

91 

67 

79 

34 

98 

188 

205 

211 

204 

160 

124 

130 

93 

45 

25 

101 


9-75 


1827 


11-33 


1828 


11-38 


1829 


14-74 


1830 


12-13 


1831 


12-22 


1832 

1833 




1834 




1835 

1836 

1837 

1838 


9-57 
12-34 
12-27 
12-74 


1839 


11-03 


1840 


9-91 


1841 • 

1842 


7-82 
7-08 


1843 


7-15 


1844 


6-61 


1845 


8-13 


1846 


8-81 


1847 


9-55 


1848 


11-15 


1849 


10-64 


1850 


10-44 


1851 


8-32 


1852 


8-09 


1853 


7-09 


1854 

1855 


6-81 
6-41 


1856 


5-98 


1857 


6-95 


1858 


7-41 


1859 


10-37 


1860 


10-05 


1861 


9-17 


1862 


8-59 


1863 


8-84 


1864 


8-02 


1865 


8 14 


1866 


7*65 


1867 . 


7 09 


1868 


8-15 







294 ASTRONOMY. 

The periodicity of the spots is evident from the table. 
It will appear in a more striking way from the following 
summary : 

From 1828 to 1831, sun without spots on only. ... 1 day. 

In 1833, " " " .... 139 days. 

From 1836 to 1840, " " " 3 " 

In 1843, " " " .... 147 " 

From 1847 to 1851, " " " 2 " 

In 1856, ° " " 193 " 

From 1858 to 1861, " " " no day. 

In 1867, " " " 195 days. 

Every 11 years there is a minimum number of spots, 
and about 5 years after each minimum there is a maxi- 
mum. If instead of merely counting the number of spots, 
measurements are made on solar photograms, as they 
are called, of the extent of spotted area, the period comes 
out with greater distinctness. This periodicity of the 
area of the solar spots appears to be connected with mag- 
netic phenomena on the earth's surface, and with the num- 
ber of auroras visible. It has been supposed to be con- 
nected also with variations of temperature, of rainfall, 
and with other meteorological phenomena such as the mon- 
soons of the Indian Ocean, etc. The cause of this period- 
icity is as yet unknown. Carrlng-ton, De la Rue, 
Loewy, and Stewart have given reasons which go to show 
that there is a connection between the spotted area and the 
configurations of the planets, particularly of Jupiter, 
Venus, and Mercury. Zollxer says that the cause lies 
within the sun itself, and assimilates it to the periodic 
action of a geyser, which seems to be a priori probable. 
Since, however, the periodic variations of the spots cor- 
respond to the magnetic variation, as exhibited in the last 
column of the table of Scfwabe's results, it appears that 
there may be some connection of an unknown nature 
between the sun and the earth at least. But at present 
we can only state our limited knowledge and wait for 
further information. 



PERIODICITY OF SUN-SPOTS. 



205 



Dr. Wolf (Director of the Zurich Observatory) has col- 
lected all the available observations of the solar spots, and 
it is found that since 1610 we have a tolerably complete 
record of these appearances. The number and character 
of the spots are now noted every day by observers in many 
quarters of the civilized world. This long series of obser- 
vations has served as a basis to determine each epoch of 
maximum and minimum which has occurred since 1610, 
and from thence to determine the length of each single 
period. 

The following table gives Dr. Wolf's results : 

Table giving the Times of Maximum and Minimum Sun-Spot 
Frequency, according to Wolf. 



First Series. 


Second Series. 


Minima. 


Diff. 


Maxima. 


Diff. 


Minima. 


Diff. 


Maxima. 


Diff. 


A.D. 1610-8 




1615-5 




1745-0 




1750-3 






8-2 




10-5 




10-2 




11-2 


1619-0 




1626-0 




1755-2 




1761-5 






15-0 




13-5 




11-3 




8-2 


1634-0 




1639-5 




1766-5 




' 1769-'; 






11-0 




9-5 




9-0 




8-7 


1645-0 




1649-0 




1775-5 




1778-4 






10-0 




11-0 




9-2 




9-7 


1655-0 




1660-0 




1784-7! 


1788-1 






11-0 




15-0 


13-6 




16-1 


1666-0 




1675-0 




1798-3 


1804-2 






13-5 




10-0 


12-3 




12-2 


1679-5 




1685-0 




1810-6 


1816-4 






10-0 




8-0 




12-7 




13-5 


1689-5 




1693-0 




1823-3 




1829-9 






8-5 




12-5 




10-6 




7-3 


1698-0 




1705-5 




1833-9 




1837-2 






14-0 




12-7 




9 6 




10-9 


1712-0 




1718-2 




1843-5 


1848-1 






11-5 




9-3 


| 12-5 




12-0 


1723-5 




1727-5 




1856-0, 


1860-1 






10-5 




11-2 




11-2 




10-5 


1734-0 




1738-7 




1867-2 




1870-1 




ll-20±2-ll 


years. 


11-20 ±2 


•06 vs. 


11 -11 ±1-54 vs. 


10-94 ±2 -52 vs. 


±0-64 




±c 


•63 


±0-47 


±0-76 



296 ASTRONOMY. 

From the first series of earlier observations, the period 
comes out from observed minima, 11-20 years, with a 
variation of two years ; from observed maxima the period 
is 11 • 20 years, with variation of three years — that is, this 
series shows the period to vary between 13-3 and 9-1 
years. If we suppose these errors to arise only from errors 
of observation, and not to be real changes of the period 
itself, the mean period is 11 • 20 ± • 64. 

The results from the second series are also given at 
the foot of the table. From a combination of the two, it 
follows that the mean period is 11-111 ± 0-307 years, 
with an oscillation of ± 2 • 030 years. 

These results are formulated by Dr. Wolf as follows : 
The frequency of solar spots has continued to change 
periodically since their discovery in 1610 ; the mean length 
of the period is 11^ years, and the separate periods may 
differ from this mean period by as much as 2-03 years. 

A general relation between the frequency of the spots and the 
variation of the magnetic needle is shown by the numbers which 
have been given in the table of Schwabe's results. This relation 
has been most closely studied by AVolf. He denotes by g the 
number of groups of spots seen on any day on the sun, counting 
each isolated spot as a group ; by/" is denoted the number of spots 
in each group (Jg is then proportional to the spotted area) ; by Jc a 
coefficient depending upon the size of the telescope used for obser- 
vation, and by r the daily relative number so called ; then he sup- 
poses 

r = 7c(f+lQ.g). 

From the daily relative numbers are formed the mean monthly 
and the mean annual relative numbers r. Then, according to 
Wolf, if v is the mean annual variation of the magnetic needle at 
any place, two constants for that place, a and j3, can be found, so- 
that the following formula is true for all years : 

v = a-\r j3-r. 
Thus for Munich the formula becomes, 
v= 6' -27 + 0'-051r; 
and for Prague, 

v — 5' -80 + 0'-045 r, and so on. 



TOTAL ECLIPSES OF THE SUN. 



297 



Year. 


Munich. 


Prague. 


Observed. 


Computed. 


A 


Observed. 


Computed. 


A 


1870 

1871 

1872 

1873 


12-27 

11-70 

10-96 

9-12 


12-77 

11-56 

11-13 

9-54 


- 0-50 
+ 0-14 
-0-17 

- 0-42 


11-41 

11-60 

10-70 

9-05 


12-10 
10-89 
10-46 

8-87 


- 0-69 
+ 0-71 
-f 0-24 
+ 0-18 



The above comparison bears out the conclusion that the 
magnetic variations are subjected to the same perturba- 
tions as the development of the solar spots, and it may 
be said that the changes in the frequency of solar spots 
and the like changes of magnetic variations show that 
these two phenomena are dependent the one on the other, 
or rather upon the same cosmical cause. What this cause 
is remains as yet unknown. 



§ 4. THE SUN'S CHROMOSPHERE AND CORONA. 

Phenomena of Total Eclipses. — The beginning of a 
total solar eclipse is an insignificant phenomenon. It is 
marked simply by the small black notch made in the lu- 
minous disk of the sun by the advancing edge or limb of 
the moon. This always occurs on the western half of the 
sun, as the moon moves from west to east in its orbit. An 
liour or more must elapse before the moon has advanced 
sufficiently far in its orbit to cover the sun's disk. During 
this time the disk of the sun is gradually hidden until it 
^becomes a thin crescent. To the general spectator there 
is little to notice during the first two thirds of this period 
from the beginning of the eclipse, unless it be perhaps the 
altered shapes of the images formed by small holes or 
apertures. Under ordinary circumstances, the image of 
the sun, made by the solar rays which pass through a small 
hole— in a card, for example— are circular in shape, like the 
shape of the sun itself. When the sun is crescent , the 



298 ASTRONOMY. 

image of the sun formed by such rays is also crescent, 
and, under favorable circumstances, as in a thick forest 
where the interstices of the leaves allow such images to be 
formed, the effect is quite striking. The reason for this 
phenomenon is obvious. 

The actual amount of the sun's light may be diminished 
to two thirds or three fourths of its ordinary amount with- 
out its being strikingly perceptible to the eye. What is 
first noticed is the change which takes place in the color 
of the surrounding landscape, which begins to wear a rud- 
dy aspect. This grows more and more pronounced, and 
gives to the adjacent country that weird effect which lends 
so much to the impressiveness of a total eclipse. The rea- 
son for the change of color is bimple. We have already 
said that the sun's atmosphere absorbs a large proportion 
of the bluer rays, and as this absorption is dependent on 
the thickness of the solar atmosphere through which the 
rays must pass, it is plain that just before the sun is total- 
ly covered the rays by which we see it will be redder than 
ordinary sunlight, as they are those which come from 
points near the sun's limb, where they have to pass through 
the greatest thickness of the sun's atmosphere. 

The color of the light becomes more and more lurid up 
to the moment when the sun has nearly disappeared. If 
the spectator is upon the top of a high mountain, he can 
then begin to see the moon's shadow rushing toward him 
at the rate of a mile in about two seconds. Just as the 
shadow reaches him there is a sudden increase of darkness 
— the brighter stars begin to shine in the dark lurid sky, 
the thin crescent of the sun breaks up into small points or 
dots of light, which suddenly disappear, and the moon it- 
self, an intensely black ball, appears to hang isolated in the 
heavens. 

An instant afterward, the corona is seen surrounding the 
black disk of the moon with a soft effulgence quite differ- 
ent from any other light known to us. Xear the moon's 
limb it is intensely bright, and to the naked eye uniform 



TOTAL ECLIPSES OF THE SUN. 299 

in structure ; 5' or 10' from the limb this inner corona 
has a boundary more or less defined, and from this extend 
streamers and wings of fainter and more nebulous light. 
These are of various shapes, sizes, and brilliancy. No 
two solar eclipses yet observed have been alike in this re- 
spect. 

These wings seem to vary from time to time, though at 
nearly every eclipse the same phenomena are described by 
observers situated at different points along the line of 
totality. That is, these appearances, though changeable, 
do not change in the time the moon's shadow requires to 
pass from Vancouver's Island to Texas, for example, which 
is some fifty minutes. 

Superposed upon these wings may be seen (sometimes 
with the naked eye) the red flames or protuberances which 
were first discovered during a solar eclipse. These need 
not be more closely described here, as they can now be 
studied at any time by aid of the spectroscope. 

The total phase lasts for a few minutes (never more than 
six or seven), and during this time, as the eye becomes more 
and more accustomed to the faint light, the outer corona is 
seen to stretch further and further away from the sun's 
limb. At the eclipse of 1878, July 29th, it was seen by 
Professor Langley, and by one of the writers, to extend 
more than 6° (about 9,000,000 miles) from the sun's limb. 
Just before the end of the total phase there is a sudden 
increase of the brightness of the sky, due to the increased 
illumination of the earth's atmosphere near the observer, 
and in a moment more the sun's rays are again visible, 
seemingly as bright as ever. From the end of totality till 
the last contact the phenomena of the first half of the 
eclipse are repeated in inverse order. 

Telescopic Aspect of the Corona. — Such are the ap- 
pearances to the naked eye. The corona, as seen through 
a telescope, is, however, of a very complicated structure. 
The inner corona is usually composed of bright stria? or fil- 
aments separated by darker bands, and some of these lat- 



300 ASTRONOMY. 

ter are sometimes seen to be almost totally black. The 
appearances are extremely irregular, but they are often as 
if the inner corona were made up of brushes of light on a 
darker background. The direction of these brushes is 
often radial to the sun, especially about the poles, but 
where the outer corona joins on to the inner these brushes 
are sometimes bent over so as to join, as it were, the 
boundaries of the outer light. 

The great difficulties in the way of studying the corona 
have been due to the short time at the disposal of the ob- 
server, and to the great differences which even the best 
draughtsmen will make in their rapid sketches of so com- 
plicated a phenomenon. The figure of the inner corona 
on the next page is a copy of one of the best drawings made 
of the eclipse of 1869, and is inserted chiefly to show the 
nature of the only drawings possible in the limited time. 
The numbers refer to the red prominences around the limb . 
The radial structure of the corona and its different exten- 
sion and nature at different points are also indicated in the 
drawing. 

The figure on page 302, is acopy of a crayon drawing made in 1878. 
The best evidence which we can gain of the details of the corona 
comes, however, from a series of photographs taken during the whole 
of totality. A photograph with a short exposure gives the details 
of the inner corona well, but is not affected by the fainter outlying 
parts. One of longer exposure shows details further away from 
the sun's limb, while those near it are lost in a glare of light, being 
over-exposed, and so on. In this way a series of photographs 
gives us the means of building up, as it were, the whole corona 
from its brightest parts near the sun's limb out to the faintest por- 
tions which will impress themselves on a photographic plate. 

The corona and red prominences are solar appendages. 
It was formerly doubtful whether the corona was an 
atmosphere belonging to the sun or to the moon. At the 
eclipse of 1860 it was proved by measurements that the 
red prominences belonged to the sun and not to the moon, 
since the moon gradually covered them by its motion, 
they remaining attached to the sun. The corona has also 
since been shown to be a solar appendage. 



TOTAL ECLIPSES OF THE SUN. 



301 



The eclipse of 1851 was total in Sweden and neigh- 
boring parts, and was very carefully observed. Similar 
prominences were seen about the sun's limb, and one of 
so bizarre a form as to show that it could by no possibility 




Fig. 82. — drawing of the corona made during the eclipse of 
august 7, 1869. 



be a mountain or solid mass, since if such had been the 
case it would inevitably have overturned. It was there- 
fore a gaseous or cloud -like appendage belonging to the 



302 



ASTRONOMY. 




Fig. 83.— sun's corona during the eclipse of july 29, 187; 



THE STTN'S PROMINENCES. 



303 



sun. There were others of various and perhaps varying 
shapes, and the "bases of these were connected by a low 
band of serrated rose-colored light. One of these protu- 
berances was shown to be entirely above the sun, as if 
floating within its atmosphere. Around the whole disk 
of the sun a ring of similar nature to the prominences 
exists, which is brighter than the corona, and seems to 
form a base for the protuberances themselves ; this is 
the sierra. Some of the red flames were of enormous 
height ; one of at least 80,000 miles. 




Fig. 84. — forms of the solar prominences as seen with the 

SPECTROSCOPE. 

G-aseous Nature of the Prominences. — The next eclipse 
(1868, July) was total in India, and was observed by many 
skilled astronomers. A discovery of M. Janssen's* will 
make this eclipse forever memorable. lie was provided 
with a spectroscope, and by it observed the prominences. 
One prominence in particular was of vast size, and when 
the spectroscope was turned upon it, its spectrum was dis- 
continuous, showing the bright lines of hydrogen gas, 

* Now Director of the Solar Observatory of Meudou, near Paris. 



304 ASTRONOMY. 

The brightness of the spectrum was so marked that 
Janssen determined to keep his spectroscope fixed npon it 
even after the reappearance of sunlight, to see how long it 
conld be followed. It was found that its spectrum could 
still be seen after the return of complete sunlight ; and not 
only on that day, but on subsequent days, similar phenom- 
ena could be observed. 

One great difficulty was conquered in an instant. The 
red flames which formerly were only to be seen for a few 
moments during the comparatively rare occurrences of 
total eclipses, and whose observation demanded long and 
expensive journeys to distant parts of the world, could 
now be regularly observed with all the facilities offered by 
a fixed observatory. 

This great step in advance was independently made by 
Mr. Lookyek,* and his discovery was derived from pure 
theory, unaided by the eclipse itself. By this method 
the prominences have been carefully mapped day by 
day all around the sun, and it has been proved that 
around this body there is a vast atmosphere of hydrogen 
gas — the chromosphere or sierra. From out of this the 
prominences are projected sometimes to heights of 100,000 
kilometres or more. 

It will be necessary to recall the main facts of observation which are 
fundamental in the use of the spectroscope. When a brilliant point is 
examined with the spectroscope, it is spread out by the prism into a 
band — the spectrum. Using two prisms, the spectrum becomes longer, 
but the light of the surface, being spread over a greater area, is en- 
feebled. Three, four, or more prisms spread out the spectrum propor- 
tionally more. If the spectrum is of an incandescent solid or liquid, it 
is always continuous, and it can be enfeebled to any degree ; so that 
any part of it can be made as feeble as desired. 

This method is precisely similar in principle to the use of the telescope 
in viewing stars in the daytime. The telescope lessens the brilliancy 
of the sky, while the disk of the star is kept of the same intensity, 
as it is a point in itself. It thus becomes visible. If it is a glowing gas, 
its spectrum will consist of a definite number of lines, say three — A, B, 
C, for example. Now suppose the spectrum of this gas to be superposed 
on the continuous spectrum of the sun ; by using only one prism, the 

* Mr. J. Norman Lockyer, F.R.S., of London, now attached to 
the Science and Art Department of the South Kensington Museum. 



THE SUN'S HEAT. 305 

solar spectrum is short and brilliant, and every part of it may be more 
brilliant than the line spectrum of the gas. By increasing the disper- 
sion (the number of prisms), the solar spectrum is proportionately en- 
feebled. If the ratio of the light of the bodies themselves, the sun and 
the gas, is not too great, the continuous spectrum may be so enfeebled 
that the line spectrum will be visible when superposed upon it, and 
the spectrum of the gas may then be seen even in the presence of true 
sunlight. Such was the process imagined and successfully carried out 
by Mr. Lockyer, and such is in essence the method of viewing the 
prominences to-day adopted. 

The Coronal Spectrum. — In 1869 (August 7th) a total solar 
eclipse was visible in the United States. It was probably observed 
by more astronomers than any preceding eclipse. Two American 
astronomers, Professor Young, of Dartmouth College, and Professor 
Harkness, of the Naval Observatory, especially observed the spec- 
trum of the corona. This spectrum was found to consist of one 
faint greenish line crossing a faint continuous spectrum. The 
place of this line in the maps of the solar spectrum published by 
Kirchhoff was occupied by a line which he had attributed to the 
iron spectrum, and which had been numbered 1474 in his list, so 
that it is now spoken of as 1474 K. This line is probably due to 
some gas which must be present in large and possibly variable 
quantities in the corona, and which is not known to us on the earth, 
in this form at least. It is probably a gas even lighter than hydro- 
gen, as the existence of this line has been traced 10' or 20' from 
the sun's limb nearly all around the disk. 

In the eclipse of July 29th, 1878, which was total in Colorado 
and Texas, the continuous spectrum of the corona was found to be 
crossed by the dark lines of the solar spectrum, showing that the 
coronal light was composed in part of reflected sunlight. 



§ 5. SOURCES OF THE SUN'S HEAT. 

Theories of the Sun's Constitution. — No considerable 
fraction of the heat radiated from the sun returns to it 
from the celestial spaces, since if it did the earth would 
intercept some of the returning rays, and the temperature 
of night would be more like that of noonday. But we 
know the sun is daily radiating into space 2,170,000,000 
times as much heat as is daily received by the earth, and 
it follows that unless the supply of heat is infinite (which 
we cannot believe), this enormous daily radiation must in 
time exhaust the supply. When the supply is exhausted, 
or even seriously trenched upon, the result to the inhab- 
itants of the earth will be fatal. A slow diminution of 



306 ASTRONOMY. 

the daily supply of heat would produce a slow change of 
climates from hotter toward colder. The serious results 
of a fall of 50° in the mean annual temperature of the 
earth will be evident when we remember that such a fall 
would change the climate of France to that of Spitzber- 
gen. The temperature of the sun cannot be kept up by 
the mere combustion of its materials. If the sun were 
solid carbon, and if a constant and adequate supply of 
oxygen were also present, it has been shown that, at the 
present rate of radiation, the heat arising from the com- 
bustion of the mass would not last more than 5000 years. 

An explanation of the solar heat and light has been 
suggested, which depends upon the fact that great amounts 
of heat and light are produced by the collision of two 
rapidly moving heavy bodies, or even by the passage of 
-a heavy body like a meteorite through the earth's atmos- 
phere. In fact, if we had a certain mass available with 
which to produce heat in the sun, and if this mass were of 
the best possible materials to produce heat by burning, 
it can be shown that, by burning it at the surface of the 
sun, we should produce vastly less heat than if we simply 
allowed it to fall into the sun. In the last case, if it fell 
from the earth's distance, it would give 6000 times more 
heat than by its burning. 

The least velocity with which a body from space could 
fall upon the sun's surface is in the neighborhood of 280 
miles in a second of time, and the velocity may be as great 
as 350 miles. From these facts, the meteoric theory of 
solar heat originated. It is in effect that the heat of the 
sun is kept up by the impact of meteors upon its surface. 

No doubt immense numbers of meteorites fall into the 
sun daily and hourly, and to each one of them a certain 
considerable portion of heat is due. It is found that, to 
account for the present amount of radiation, meteorites 
equal in mass to the whole earth would have to fall into 
the sun every century. It is extremely improbable that a 
mass one tenth as large as this is added to the sun in this 



SUPPLY OF SOLAR IIEAT. 307 

way per century, if for no other reason because the earth 
itself and every planet would receive far more than its 
present share of meteorites, and would itself become quite 
hot from this cause alone. 

There is still another way of accounting for the sun's 
constant supply of energy, and this has the advantage of 
appealing to no cause outside of the sun itself in the ex- 
planation. It is by supposing the heat, light, etc. , to be 
generated by a constant and gradual contraction of the 
dimensions of the solar sphere. As the globe cools by 
radiation into space, it must contract. In so contracting its 
ultimate constituent parts are drawn nearer together by 
their mutual attraction, whereby a form of energy is de- 
veloped which can be transformed into heat, light, elec- 
"tricity, or other physical forces. 

This theory is in complete agreement with the known 
laws of force. It also admits of precise comparison with 
facts, since the laws of heat enable us, from the known 
amount of heat radiated, to infer the exact amount of con- 
traction in inches which the linear dimensions of the sun 
must undergo in order that this supply of heat may be 
kept unchanged, as it is practically found to be. With 
ihe present size of the sun, it is found that it is only 
necessary to suppose that its diameter is diminishing at the 
Tate of about 220 feet per year, or 4 miles per century, 
in order that the supply of heat radiated shall be constant. 
It is plain that such a change as this may be taking place, 
since we possess no instruments sufficiently delicate to 
have detected a change of even ten times this amount 
since the invention of the telescope. 

It may seem a paradoxical conclusion that the cooling 
of a body may cause it to become hotter. This indeed is 
true only when we suppose the interior to be gaseous, and 
not solid or liquid. It is, however, proved by theory that 
this law holds for gaseous masses. 

If a spherical mass of gas be condensed to one half the primitive 
diameter, the central attraction upon any part of its mass will be in- 



308 ASTRONOMY. 

creased fourfold, while the surface subjected to this attraction will 
be reduced to one fourth. Hence the pressure per unit of surface 
will be augmented sixteen times, while the density will be increased 
but eight times. If the elastic and the gravitating forces were in 
equilibrium in the original condition of the mass, the temperature 
must be doubled in order that they may still be in equilibrium when 
the diameter is reduced to one half. 

If, however, the primitive body is originally solid or liquid, or if, 
in the course of time, it becomes so, then this law ceases to hold, and 
radiation of heat produces a lowering of the temperature of the 
body, which progressively continues until it is finally reduced to the 
temperature of surrounding space. 

We cannot say whether the sun has yet begun to liquefy 
in his interior parts, and hence it is impossible to predict 
at present the duration of his constant radiation. Theory 
shows us that after about 5,000,000 years, the sun radiating- 
heat as at present, and still remaining gaseous, will be re- 
duced to one half of its present volume. It seems prob- 
able that somewhere about this time the solidification 
will have begun, and it is roughly estimated, from this 
line of argument, that the present conditions of heat radi- 
ation cannot last greatly over 10,000,000 years. 

The future of the sun (and hence of the earth) cannot, 
as we see, be traced with great exactitude. The past can 
be more closely followed if we assume (which is tolerably 
safe) that the sun up to the present has been a gaseous, and 
not a solid or liquid mass. Four hundred years ago, 
then, the sun was about 16 miles greater in diameter 
than now ; and if we suppose this process of contrac- 
tion to have regularly gone on at the same rate (an 
uncertain supposition), we can fix a date when the sun 
filled any given space, out even to the orbit of Nep- 
tune — that is, to the time when the solar system consisted 
of but one body, and that a gaseous or nebulous one. 
It will subsequently be seen that the ideas here reached 
a posteriori have a striking analogy to the a priori ideas 
of Kant and La Place. 

It is not to be taken for granted, however, that the 
amount of heat to be derived from the contraction of the 



AGE OF THE 8 UN. 309 

sun's dimensions is infinite, no matter how large the prim- 
itive dimensions may have been. A body falling from 
any distance to the sun can only have a certain finite veloc- 
ity depending on this distance and the mass of the sun 
itself, which, even if the fall be from an infinite distance, 
cannot exceed, for the sun, 350 miles per second. In 
the same way the amount of heat generated by the con- 
traction of the sun's volume from any size to any other is 
finite, and not infinite. 

It has been shown that if the sun has always been 
radiating heat at its present rate, and if it had originally 
filled all space, it has required 18,000,000 years to contract 
to its present volume. In other words, assuming the pres- 
ent rate of radiation, and taking the most favorable case, 
the age of the sun does not exceed 18,000,000 years. The 
earth, is of course, less aged. The supposition lying at the 
base of this estimate is that the radiation of the sun has 
been constant throughout the whole period. This is quite 
unlikely, and any changes in this datum affect greatly the 
final number of years which we have assigned. While 
this number may be greatly in error, yet the method of 
obtaining it seems, in the present state of science, to be 
satisfactory, and the main conclusion remains that the past 
of the sun is finite, and that in all probability its future is 
a limited one. The exact number of centuries that it is to 
last are of no moment even were the data at hand to ob- 
tain them : the essential point is, that, so far as we can 
see, the sun, and incidentally the solar system, has a finite 
past and a limited future, and that, like other natural ob- 
jects, it passes through its regular stages of birth, vigor, 
decay, and death, in one order of progress. 



CHAPTER III. 

THE INFERIOR PLANETS. 

§ 1. MOTIONS AND ASPECTS. 

The inferior planets are those whose orbits lie between 
the sun and the orbit of the earth. Commencing with the 
more distant ones, they comprise Venus, Mercury, and, in 
the opinion of some astronomers, a planet called Vulcan, 
or a group of planets, inside the orbit of Mercury. The 
planets Mercury and Venus have so much in common that 
a large part of what we have to say of one can be applied 
to the other with but little modification. 

The real and apparent motions of these planets have 
already been briefly described in Part I. , Chapter TV. It 
will be remembered that, in accordance with Kepler' & 
third law, their periods of revolution around the sun are 
less than that of the earth. Consequently they overtake 
the latter between successive inferior conjunctions. 

The interval between these conjunctions is about four 
months in the case of Mercury, and between nineteen and 
twenty months in that of Venus. At the end of this 
period each repeats the same series of motions relative to 
the sun. What these motions are can be readily seen by 
studying Fig. 84. In the first place, suppose the earth, 
at any point, E, of its orbit, and if we draw a line, E L 
or EM, from E, tangent to the orbit of either of these 
planets, it is evident that the angle which this line makes 
with that drawn to the sun is the greatest elongation of 
the planet from the sun. The orbits being eccentric, this 



ASPECTS OF MERCURY AND VENUS. 



311 



elongation varies with the position of the earth. In the 
case of Mercury it ranges from 16° to 29°, while in the 
case of Venus, the orbit of which is nearly circular, it 

varies very little from 
45°. These planets, 
therefore, seem to have 
an oscillating motion, 
first swinging toward the 
east of the sun, and then 
toward the west of it, as 
already explained in Part 
I., Chapter IY. Since, 
owing to the annual revo- 
lution of the earth, the 
sun has a constant east- 
ward motion among the 
stars, these planets must 
have, on the whole, a corresponding though intermittent 
motion in the same direction. Therefore the ancient 
astronomers supposed their period of revolution to be one 




Fig. 84. 



year, 



the same as that of the sun. 



If, again, we draw a line JES C from the earth through 
the sun, it is evident Uiat the first point 7, in which this 
line cuts the orbit of the planet, or the point of inferior 
conjunction, will (leaving eccentricity out of the question) 
be the least distance of the planet from the earth, while the 
second point (7, or the point of 
superior conjunction, on the op- 
posite side of the sun, will be 
the greatest distance. Owing to 
the difference of these distances, 
the apparent magnitude of these 
planets, as seen from the earth, 
is subject to great variations. 

Fig. 85 shows these variations in the case of Mercury, 
A representing its apparent magnitude when at its greatest 
distance, B when at its mean distance, and G when at its 




Fig. 85. — apparent magni- 
tudes OF THE DISK OF 
MERCURY. 



^^^^■^^^^M 



312 ASTRONOMY. 

least distance. In the case of Venus (Fig. 86) the varia* 
tions are much greater than in that of Mercury, the great- 
est distance, 1-72, being more than six times the least 
distance, which is only • 28. The variations of apparent 
magnitude are therefore great in the same proportion. 

In thus representing the apparent angular magnitude 
of these planets, we suppose their whole disks to be visible, 
as they would be if they shone by their own light. But 
since they can be seen only by the reflected light of the 
sun, only those portions of the disk can be seen which are 
at the same time visible from the sun and from the earth. 
A very little consideration will show that the proportion 
of the disk which can be seen constantly diminishes as the 
planet approaches the earth, and looks larger. 




Pig. 86. — appabent magnitudes op disk op venus. 

"When the planet is at its greatest distance, or in superior 
conjunction ((7, Fig. 84), its whole illuminated hemisphere 
can be seen from the earth. As it moves around and ap- 
proaches the earth, the illuminated hemisphere is gradually 
turned from us. At the point of greatest elongation, M 
or L, one half the hemisphere is visible, and the planet 
has the form of the moon at first or second quarter. As 
it approaches inferior conjunction, the apparent visible disk 
assumes the form of a crescent, which becomes thinner 
and thinner as the planet approaches the sun. 

Fig. 87 shows the apparent disk of Mercury at various 
times during its synodic revolution. The planet will ap- 
pear brightest when this disk has the greatest surface. 



ASPECTS OF MERCURY AND VENUS. 



313 



This occurs about half way between greatest elongation 
uid inferior conjunction. 

In consequence of the changes in the brilliancy of these 
planets produced by the variations of distance, and those 
produced by the variations in the proportion of illuminated 
disk visible from the earth, partially compensating each 
other, their actual brilliancy is not subject to such great 
variations as might have been expected. As a general rule, 
Mercury shines with a light exceeding that of a star of 
the first magnitude. But owing to its proximity to the 
sun, it can never be seen by the naked eye except in the 
west a short time after sunset, and in the east a little be- 
fore sunrise. It is then of necessity near the horizon, and 



• I ) ) 



B C 






Fig. 87. — appearance of mercury at different points of its 

ORBIT. 

therefore does not seem so bright as if it were at a greater 
altitude. In our latitudes we might almost say that it is 
never visible except in the morning or evening twilight. 
In higher latitudes, or in regions where the air is less 
transparent, it is scarcely ever visible without a telescope. 
It is said that Copernicus died without ever obtaining a 
view of the planet Mercury. 

On the other hand, the planet Venus is, next to the sun 
and moon, the most brilliant object in the heavens. It is 
so much brighter than any fixed star that there can seldom 
be any difficulty in identifying it. The unpractised ob- 
server might under some circumstances find a difficulty 1Q 



314 ASTRONOMY. 

distinguishing between Venus and Jupiter, but the differ- 
ent motions of the two planets will enable him to distin- 
guish them if they are watched from night to night dur- 
ing several weeks. 

§ 2. ASPECT AND ROTATION OF MERCURY. 

The various phases of Mercury, as dependent upon its 
various positions relative to the sun, have already been 
shown. If the planet were an opaque sphere, without in- 
equalities and without an atmosphere, the apparent disk 
would always be bounded by a circle on one side and an 
ellipse on the other, as represented in the figure. 
"Whether any variation from this simple and perfect form 
has ever been detected is an open question, the balance of 
evidence being very strongly in the negative. Since no 
spots are visible upon it, it would follow that unless vari- 
ations of form due to inequalities on its surface, such as 
mountains, can be detected, it is impossible to determine 
whether the planet rotates on its axis. The only evidence 
in favor of such rotation is that of Schroter, the celebrated 
astronomer of Lilienthal, who made the telescopic study 
of the moon and planets his principal work. About the 
beginning of the present century he noticed that at certain 
times the south horn of the crescent of Mercury seemed 
to be blunted. Attributing this appearance to the shadow 
of a lofty mountain, he concluded that the planet Mercury 
revolved on its axis in a little more than 24 hours. But 
this planet has since been studied with instruments much 
more powerful than those of Schroter, and no confirma- 
tion of his results has been obtained. "We must therefore 
conclude that the period of rotation of Mercury on its 
axis is entirely unknown. 

Respecting an atmosphere of Mercury, the evidence is 
also conflicting. The spectrum of this planet has been 
studied by Dr. Yogel, now astronomer at the Physical 
Observatory of Potsdam, who finds that its principal lines 



ASPECTS OF MERCURY 315 

coincide with those of the sun. Of course we should 
expect this because the planet shines by reflected solar 
light. But he also linds that certain lines are seen in the 
spectrum of Mercury which we know to be due to the ab- 
sorption of the earth's atmosphere, and which appear 
more dense than they should from the simple passage 
through our atmosphere. This would seem to show that 
Mercury has an envelope of gaseous matter somewhat like 
our own. On the other hand, Dr. Zollner, of Leipsic, 
by measuring the amount of light reflected by the planet 
at various times, concludes that Mercury, like our moon, 
is devoid of any atmosphere sufficient to reflect the light 
of the sun. We may therefore regard it as doubtful 
whether any evidence of an atmosphere of Mercury can 
be obtained, and it is certain that we know nothing defi- 
nite respecting its physical constitution. 



§ 3. THE ASPECT AND SUPPOSED ROTATION OF 

VENUS. 

As Venus sometimes comes nearer the earth than any 
other primary planet, astronomers have examined its sur- 
face with great interest ever since the invention of the 
telescope. But no conclusive evidence respecting the ro- 
tation of the planet and no proof of any changes or any 
inequalities on its surface have ever been obtained. The 
observations are either very discordant, or so difficult 
and unreliable that we may readily suppose the ob- 
servers to have been misled as to what they saw. In 1767 
Cassini thought he saw a bright spot on Venus during 
several successive evenings, and concluded, from his sup- 
posed observation that the planet revolved on its axis in a 
little more than 23 hours. The subject was next taken up 
by Blanchini, an Italian astronomer, who supposed that 
he saw a number of dark regions on the planet. These he 
considered to be seas or oceans, and he went so far as to 
give them names. Watching them from night to night, 



316 ASTRONOMY. 

he concluded that the time of rotation of Venus was more 
than 24 days. Again, Schroter thought that, when Ve- 
nus was a crescent, one of its sharp points was blunted 
at certain intervals, as in the case of Mercury. He formed 
the same theory of the cause of this appearance — namely, 
that it was due to the shadow of a high mountain. He con- 
cluded that the time of rotation found by Cassini was near- 
ly correct. Finally, in 1842, De Yico, of Rome, thought 
he could see the same dark regions or oceans on the planet 
which had been seen by Blanchini. He concluded that the 
true time of rotation was 23 h 21 m 22 s . This result has gone 
into many of our text-books as conclusive, but it is contra- 
dicted by the investigation of many excellent observers 
with much better instruments. Herschel was never able to 
see any permanent markings on Venus. If he ever caught 
a glimpse of spots, they were so transient that he could 
gather no evidence respecting the rotation of the planet. 
He therefore concluded that if they really existed, they 
w T ere due entirely to clouds floating in an atmosphere, and 
that no time of rotation could be deduced by observing 
them. This view of Herschel, so far as concerns the 
aspect of the planet, is continued by a study with the most 
powerful telescopes in recent times. With the great 
Washington telescope, no permanent dark spots and no 
regular blunting of either horn has ever been observed. 

It may seem curious that skilled observers could have 
been deceived as to what they saw ; but we must remem- 
ber that there are many celestial phenomena which are ex- 
tremely difficult to make out. By looking at a drawing 
of a planet or nebula, and seeing how jDlain every thing 
seems in the picture, we may be entirely deceived as to the 
actual aspect with a telescope. Under the circumstances, if 
the observer has any preconceived theory, it is very easy 
for him to think he sees every thing in accordance with 
that theory. ^ ow, there are at all times great differences 
in the brilliancy of the different parts of the disk of Venus, 
It is brightest near the round edge which is turned 



ASPECTS OF VENUS. 317 

toward the sun. Over a small space the brightness is such 
that some recent observers have formed a theory that the 
sun's light is reflected as from a mirror. On the other 
hand, near the . boundary between light and darkness, the 
surface is much darker. Moreover, owing to the undu- 
lations of our atmosphere, the aspect of any planet so small 
and bright as Venus is constantly changing. The only 
way to reach any certain conclusion respecting its ap- 
pearance is to take an average, as it were, of the appear- 
ances as modified by the undulations. In taking this aver- 
age, it is very easy to imagine variations of light and dark- 
ness which have no real existence ; it is not, therefore, sur- 
prising that one astronomer should follow in the footsteps 
of another in seeing imaginary markings. 

Atmosphere of Venus. — The evidence of an atmosphere 
of Venus is perhaps more conclusive than in the case of 
any other planet. When Venus is observed very near 
its inferior conjunction, and when it therefore presents the 
view of a very thin crescent, it is found that this crescent 
extends over more than 180°. This would be evidently 
impossible unless the sun illuminated more than one half 
the planet. One of the most fortunate observers of this 
phenomenon was Prof essor C. S. Lyman, of Yale College, 
who observed Venus in December, 1866. The inferior 
conjunction of the planet occurred near the ascending 
node, so that its angular distance from the sun was less 
than it had been at any former time during the present cen- 
tury. Professor Lyman saw the disk, not as a thin cres- 
cent, but as an entire and extremely fine circle of light. 
We therefore conclude that Venus has an atmosphere 
which exercises so powerful a refraction upon the light of 
the sun that the latter illuminates several degrees more 
than one half the globe. A phenomenon which must be 
attributed to the same cause has several times been ob- 
served during transits of Venus. During the transit of 
December 8th, 1874, most of the observers who enjoyed 
a fine steady atmosphere saw that when Vt m us was par- 



318 ASTRONOMY. 

tially projected on the sun, the outline of that part of its 
disk outside the sun could be distinguished by a delicate 
line of light. A similar appearance was noticed by David 
Rittenhouse, of Philadelphia, on June 3d, 1769. From 
these several observations, it would seem that the refractive 
power of the atmosphere of Venus is greater than that of 
the earth. Attempts have been made to determine its ex- 
act amount, but they are too uncertain to be worthy of 
quotation. 

§ 4. TRANSITS OF MERCURY AND VENTTS. 

When Mercury or Venus passes between the earth and 
sun, so as to appear projected on the sun's disk, the phe- 
nomenon is called a transit. If these jjlanets moved around 
the sun in the plane of the ecliptic, it is evident that 
there would be a transit at every inferior conjunction. But 
since their orbits are in reality inclined to the ecliptic, 
transits can occur only when the inferior conjunction takes 
place near the node. In order that there may be a transit, 
the latitude of the planet, as seen from the earth, must 
be less than the angular semi- diameter of the sun — that is, 
less than 16'.* 

The longitude of the descending node of Mercury at the 
present time is 227°, and therefore that of the ascending 
node 47°. The earth has these longitudes on May 7th and 
November 9th. Since a transit can occur only within a 
few degrees of a node, Mercury can transit only within a 
few days of these epochs. 

The longitude of the descending node of Venus is now 

* The mathematical student, knowing that the inclination of the orbit 
of Mercury is 7° 0' and that of Venus 3° 24', will find it an interesting 
problem to calculate the limits of distance from the node within which in- 
ferior conjunction must take place in order that a transit may occur. 
From the geocentric latitude 15 the heliocentric latitude may be found 
by multiplying by the distance from the earth and dividing by that from 
the sun. He will find these limits to be a little greater for Mercury 
than for Venus, notwithstanding its greater inclination, and to be only 
a few degrees in either qase. 



TRANSITS OF MEECVBY. 319 

about 256°, and therefore that of the ascending node is 
76°. The earth has these longitudes on June 6th and De- 
cember 7th of each year. Transits of Venus can there- 
fore occur only within two or three days of these times. 

Recurrence of Transits of Mercury.— The transits of Mer- 
cury and Venus recur in cycles which resemble the eighteen- 
year cycle of eclipses, but in which the precision of the recurrence 
is less striking. From the mean motions of Mercury and the earth 
already given, we find that the mean synodic period of Mercury is, 
in decimals of a Julian year, y - 317256. Three synodic periods are 
therefore some eighteen days less than a year. If, then, we suppose 
an inferior conjunction of Mercury to occur exactly at a node, the 
third conjunction following will take place about eighteen days 
"before the earth again reaches. the node, and therefore about 18° 
from the node, since the earth moves nearly 1° in a day. This is 
far outside the limit of a transit ; we must, therefore, wait until 
another conjunction occurs near the same place. To find when 
this will be, the successive vulgar fractions which converge toward 
the value of the above period may be found by the method of con- 
tinued fractions. The first six of these fractions are : 

JL _6_ _7_ 13. _4S 48 

3 19 22 IT 10T 14T 

Here the denominators are numbers of synodic periods, while the 
numerators are the approximate corresponding number of years. 
By actual multiplication we find : 

3 Periods = 0^-951768 = 1* —> 048232. Error = - 17° 



19 " 


= 6-027864= 6 + • 027864. 


" = + 10° 


22 " 


= 6-979632= 7- -020368. 


ci ryo 


41 " 


= 13-007496= 13+ -007496. 


" = + 2°-7 


104 " 


= 32-994618= 33-0-005382. 


" = _ l°.94 


145 " 


= 46-002120= 46+- 002120. 


" = + 0°.76 



In this table the errors show the number of degrees from the 
node at which the inferior conjunction will occur at the end of one 
year, six years, seven years, etc. They are found by multiplying 
the fraction by which the intervals exceed or fall short of an entire 
number of years by 360°. It will be seen that the 19th, 22d, 41st, 
and 145th conjunctions occur nearer and nearer the node, or, sup- 
posing that we do not start from a node, nearer and nearer the point 
of the orbits from which we do start. It follows that the recur- 
rence of a transit of Mercury at the same node is possible at the 
end of 7 years, probable at the end of 13 years, and almost certain 
at the end of 46 years. The latter is the cycle which it would be 
most convenient to take as that in which all the transits would 
recur, but it would still not be so exact as the eclipse cycle of IS 
years 11 days. 



320 



ASTRONOMY. 



The following table shows the dates of occurrence of transits of 
Mercury during the present century. They are separated into May 
transits, which occur near the descending node, and November 
ones, which occur near the ascending node. November transits are 
the most numerous, because Mercury is then nearer the sun, and 
the transit limits are wider. 



1799, May 6. 
1832, May 5. 
1845, May 8. 
1878, May 6. 
1891, May 9. 



1802, Nov. 9. 
1815, Nov. 11. 
1822, Nov. 5. 
1835, Nov. 7. 
1848, Nov. 10. 
1861, Nov. 12. 
1868, Nov. 5. 
1881, Nov. 7. 
1894, Nov. 10. 

It will be seen that in a cycle of 46 years there are two May tran- 
sits and four November ones, so that the latter are twice as nu- 
merous as the former. These numbers may, however, change slightly 
at some future time through the failure of a recurrence, or the en- 
trance of a new transit into the series. Thus, in the May series, it 
is doubtful whether there will be an actual transit 46 years after 
1891— that is, in 1937 — or whether Mercury will only pass very near 
the limb of the sun. On the other hand, Mercury passed within a few 
minutes of the sun's limb on May 3d, 1865, and it will probably 
graze the limb 46 years later — that is, on May 4th or 5th, 1911. 

Recurrence of Transits of Venus.— For many centuries 
past and to come, transits of Venus occur in a cycle more exact than 
those of Mercury. It happens that eight times the mean motion of 
Venus is very nearly the same as thirteen times the mean motion 

of the earth ; in other words, Venus 
makes 13 revolutions around the 
sun in nearly the same time that 
the earth makes 8 revolutions — 
that is, in eight years. During 
this period there w 7 ill be 5 inferior 
conjunctions of Venus, because the 
latter has made 5 revolutions more 
than the earth. Consequently, if 
we w T ait eight years from an inferior 
conjunction of Venus, we shall, at 
the end of that time, have another 
inferior conjunction, the fifth in 
regular order, at nearly the same 
point of the two orbits. It will, 
therefore, occur at the same time 
of the year, and in nearly the same 
position relative to the node of Venus. In Fig. 88 let S represent 
the sun, and the circle drawn around it the orbit of the earth. 




Fig. 88.- 



-CONJTJNCTIONS OF 
VENUS. 



TRANSITS OF VENUS. 321 

Suppose also that at the moment of the inferior conjunction of 
Venus, we draw a straight line S 1 through Venus to the earth at 1. 
We shall then have to wait about If years for another inferior con- 
junction, during which time the earth will have made one revolu- 
tion and f of another, and Venus 2f revolutions. The straight line 
drawn through the point of inferior conjunction will then be S 2. 
The third conjunction will in the same way take place in the posi- 
tion S 3, which is If revolutions further advanced ; the fourth in 
the position S 4, and the fifth in the position S 5. If the corre- 
spondence of the motions were exact, the sixth conjunction, at the 
end of 8 years (5 x If = 8), would again take place in the original 
position S 1, and all subsequent ones would follow in the same 
order. All inferior conjunctions would then take place at one of 
these five points, and no transit would ever be possible unless one 
of these points should chance to be very near the line of nodes. 

In fact, however, the correspondence is not perfectly exact, but, 
at the end of 8 years, the sixth conjunction will take place not 
exactly along the line SI, but a little before the two bodies reach 
this line. The actual angle between the line Si and that of the 
sixth conjunction will be about 2° 22', the point shifting back to- 
ward the direction #4. Of course, each following conjunction will 
take place at the same distance back from that of eight years before, 
leaving oui; small changes due to the eccentricities of the orbits and 
the variations of their elements. It follows then that if we suppose 
the five lines of conjunction to have a retrograde motion in a 
direction the opposite of that of the arrow, amounting to 2° 22' in 
eight years, all the inferior conjunctions will take place along these 
five lines. The distance apart of the lines being 72° and the 
motion about 18' per year, the intervals between the passages of 
the several conjunction lines over the line of nodes will be about 
240 years. Really, the exact time is 243 years. 

Suppose, now, that a conjunction should take place exactly at a 
node, then the fifth following conjunction would take place 
2° 22' before reaching the node. The limits within which a transit 
can occur are, however, only 1° 46' on each side of the node ; con- 
sequently, there would be no further transit at that node until the 
next following con junction point reached it, which would happen at 
the end of 243 years. If, however, the conjunction should take place 
between 0° 36' and 1° 46' after reaching the node, there would be a 
transit, and the fifth following conjunction would also occur within 
the limit on the other side of the node, so that we should have two 
transits eight years apart. We may, therefore, have either one 
transit or two according to the distance from the node at which the 
first transit occurs. We thus have at any one node either a single 
transit, or a pair of transits eight years apart, in a cycle of 243 years. 
At the middle of this cycle the node will be half way between two 
of the conjunction points — the points 1 and 3, for instance ; but it is 
evident that in this case the opposite node will coincide with the 
conjunction point 2, since there is an odd number of such points. 
It follows, therefore, that about the middle of the interval between 
two consecutive sets of transits at one node we shall have a transit 
or a pair of transits at the opposite node. 



322 ASTRONOMY. 

The earth passes through the line of the descending node of the: 
orbit of Venus early in June of each year, and through the ascending 
node early in December. It follows, therefore, that the series will 
be a transit or a pair of transits in June ; then an interval of about 120 
years, to be followed by a transit or a pair of transits in December, 
and so on. Owing to the eccentricity of the orbits, the inter vals- 
will not be exactly equal, the motions of the several conjunction 
points not being uniform, nor their distance exactly 72°. The 
dates and intervals of the transits for three cycles nearest to the 
present time are as follows : 

Intervals. 
1518, June 2. 1761, June 5. 2004, June 8. 8 years.. 

1526, June 1. 1769, June 3. 2012, June 6. 105* " 

1631, Dec. 7. 1874, Dec. 9. 2117, Dec. 11. 8 " 

1639, Dec. 4. 1882, Dec. 6. 2125, Dec. 8. 121| " 

The 243-year cycle is so exact that the actual deviations from it 
are due almost entirely to the secular variation of the orbits of 
Venus and the Earth. Moreover, the conjunction of December 8th, 
1874, took place 1° 25' past the ascending node, so that the con- 
junction of 1882 takes place about 1° 4' before reaching the node. 
Owing to the near approach of the period to exactness, several pairs 
of transits near this node have taken place in the past, at equal in- 
tervals of 243 years, and will be repeated for three or four cycles in 
the future. 

Nearly the same remark applies to those which take place at the 
descending node, where pairs of transits eight years apart will 
occur for about three cycles in the future. Owing, however, to 
secular variations of the orbit, the conjunction point for the second 
June transit of each pair and the first December transit will, after 
perhaps a thousand years, take place so far from the node that the 
planet will not quite touch the sun, and then during a period of 
many centuries there will only be one transit at each node hx 
every 243 years, instead of two, as at present. 



§ 5. SUPPOSED IWTRAMERCURIAL PLANETS. 

Some astronomers are of opinion that there is a small 
planet or a group of planets revolving around the sun 
inside the orbit of Mercury. To this supposed planet the 
name Vulcan has been given ; but astronomers generally 
discredit the existence of such a planet of considerable 
size, because the evidence in its favor is not regarded as 
conclusive. 



THE SUPPOSED VULCAN. 32S 

The evidence in favor of the existence of such planets may be 
divided into three classes, as follows, which will be considered in 
their order : 

(1) A motion of the perihelion of the orbit of Mercury, supposed 
to be due to the attraction of such a planet or group of planets. 

(2) Transits of dark bodies across the disk of the sun which have 
been supposed to be seen by various observers during/ the past cen- 
tury. 

(3) The observation of certain unidentified objects by Professor 
Watson and Mr. Lewis Swift during the total eclipse of the sun, 
July 29th, 1878. 

(1) In 1858, Le Verrier made a careful collection of all the obser- 
vations on the transits of Mercury which had been recorded since the 
invention of the telescope. The result of that investigation was 
that the observed times of transit could not be reconciled with the 
calculated motion of the planet, as due to the gravitation of the 
other bodies of the solar system. He found, however, that if, in 
addition to the changes of the orbit due to the attraction of the 
other planets, he supposed a motion 'of the perihelion amounting to 
36" in a century, the observations could all be satisfied. Such 
a motion might be produced by the attraction of an unknown 
planet inside the orbit of Mercury. Since, however, a single 
planet, in order to produce this effect, would have to be of consid- 
erable size, and since no such object had ever been observed during 
a total eclipse of the sun, he concluded that there was probably a 
group of planets much too small to be separately distinguished. 
So far as the discrepancy between theory and observation is con- 
cerned, these results of Le Verrier's have been completely con- 
firmed by the mathematical researches of Mr. G. W. Hill, and by 
observations of transits since Le Verrier's calculations were com- 
pleted. Indeed, the result of these researches and observations is- 
that the motion of the perihelion is even greater than that found 
by Le Verrier, the surplus motion being more than 40" in a cen- 
tury. There is no known way of accounting for this motion in 
accordance with well-established laws, except by supposing matter 
of some sort to be revolving around the sun in the supposed posi- 
tion. At the same time it is always possible that the effect may 
be produced by some unknown cause.* 

(2) Astronomical records contain upward of twenty instances 
in which dark bodies have been supposed to be seen in transit 
across the disk of the sun. If we suppose these observations to be 
all perfectly correct, the existence of a great number of considerable 
planets within the orbit of Mercury would be placed beyond doubt. 
But a critical analysis shows that these observations, considered as a 
class, are not entitled to the slightest credence. In the first place, 

* An electro-dynamic theory of attraction has been within the past 
twenty years suggested by several German physicists, which involves 
a small variation from the ordinary theory of gravitation. It has been 
shown that, by supposing this theory true, the motion of the perihelion 
of Mercury could be accounted for by the attraction of the sun. 



324 ASTRONOMY. 

scarcely any of them were made by experienced observers with 
powerful instruments. It is very easy for an unpractised observer 
to mistake a round solar spot for a planet in transit. It may there- 
fore be supposed that in many cases the observer saw nothing but 
a spot on the sun. In fact, the very last instance of the kind on 
record was an observation by Weber at Peckeloh, on April 4th, 
1876. He published an account of his observation, which he sup- 
posed was that of a planet, but when the publication reached other 
observers, who had been examining the sun at the same time, it 
was shown conclusively that what he saw was nothing more than 
an unusually round solar spot. Again, in most of the cases referred 
to, the object seen was described as of such magnitude that it 
could not fail to have been noticed during total eclipses if it had 
any real existence. It is also to be noted that if such planets ex- 
isted they would frequently pass over the disk of the sun. Dur- 
ing the past fifty years the sun has been observed almost every 
day with the greatest assiduity by eminent observers, armed with 
powerful instruments, who have made the study of the sun's sur- 
face and spots the principal work of their lives. None of these 
observers has ever recorded the transit of an unknown planet. This 
evidence, though negative in form, is, under the circumstances, con- 
clusive against the existence of such a planet of such magnitude 
as to be visible in transit with ordinary instruments 

(3) The observations of Professor Watson during the total 
eclipse above mentioned seem to afford the strongest evidence yet 
obtained in favor of the real existence of the planet. His mode of 
proceeding w T as briefly this : Sweeping to the west of the sun 
during the eclipse, he saw two objects in positions where, suppos- 
ing the pointing of his telescope accurately known, no fixed star 
existed. There is, however, a pair of known stars, one of which is 
about a degree distant from one of the unknown objects, and the 
other about the same distance and direction from the second. It 
is considered by some that Professor Watson's supposed planets 
may have been this pair of stars. Still, if Professor Watson's 
planets were capable of producing the motion of the perihelion of 
Mercury already referred to, we should regard their existence as 
placed beyond reasonable doubt. Eut his observations and the 
theoretical results of Le Verrier do not in any manner strengthen 
each other, because, if we suppose the observed perturbations in 
the orbit of Mercury to be due to planets so small as those seen by 
Watson, the number of these planets must be many thousands. 
Now, it is very certain that there are not thousands of planets 
there brighter than the sixth magnitude, because they would have 
been seen by other telescopes engaged in the same search. The 
smaller we suppose the individual planets, the more numerous they 
must be, and, finally, if we consider them as individually invisible, 
they will probably be numbered by tens of thousands. The smaller 
and more numerous they are, supposing their combined mass the 
same, the greater the sum total of light they would reflect. At a 
certain point the amount of light would become so considerable 
that the group would appear as a cloud-like mass. Now, there is 



THE SUPPOSED VULCAN. 325 

a phenomenon known as the zodiacal light, which is probably caused 
by matter either in a gaseous state or composed of small particles re- 
volving around the sun at various distances from it. This light 
can be seen rising like a pillar from the western horizon on any 
very clear night in the winter or spring. Of its nature scarcely 
any thing is yet known. The spectroscopic observations of Pro- 
fessor Wright, of Yale College, seem to indicate that it is seen by 
reflected sunlight. Very different views, however, have obtained 
respecting its constitution, and even its position, some having held 
that it is a ring surrounding the earth. "We can therefore merely 
suggest the possibility that the observed motion of the perihelion 
of Mercury is produced by the attraction of this mass. 



CHAPTER IV. 

THE MOON. 

In Chapter VII. of the preceding part we have de- 
scribed the motions of the moon and its relation to the 
earth. We shall now explain its physical constitution as 
revealed by the telescope. 

"When it became clearly understood that the earth and 
moon were to be regarded as bodies of one class, and that 
the old notion of an impassable gulf between the character 
of bodies celestial and bodies terrestrial was unfounded, 
the question whether the moon was like the earth in all its 
details became one of great interest. The point of most 
especial interest was whether the moon could, like the 
earth, be peopled by intelligent inhabitants. Accordingly, 
when the telescope was invented by Galileo, one of the 
first objects examined was the moon. With every im- 
provement of the instrument, the examination became 
more thorough, so that the moon has been an object of 
careful study by the physical astronomer. 

The immediate successors of Galileo thought that they 
perceived the surface of the moon, like that of our globe, 
to be diversified with land and water. Certain regions ap- 
peared dark and, for the most part, smooth, while others 
were bright and evidently broken up into hills and valleys. 
The former regions were supposed to be oceans, and re- 
ceived names to correspond with this idea. These names 
continue to the present day, although we now know that 
there are no oceans there. 

With every improvement in the means of research, it 



THE MOON. 327 

has become more and more evident that the surface of the 
moon is totally unlike that of our earth. There are no 
oceans, seas, rivers, air, clouds, or vapor. We can hardly 
suppose that animal or vegetable life exists under such 
circumstances, the fundamental conditions of such ex- 
istence on our earth being entirely wanting. We might 
almost as well suppose a piece of granite or lava to be the 
abode of life as the surface of the moon to be such. 

Before proceeding with a description of the lunar sur- 
face, as made known to us by the telescopes of the present 
time, it will be well to give some estimates of the visi- 
bility of objects on the moon by means of our instruments. 
Speaking in a rough way, we may say that the length of 
one mile on the moon would, as seen from the earth, sub- 
tend an angle of \" of arc. More exactly, the angle sub- 
tended would range between 0"-8 and 0"-9, according to 
the varying distance of the moon. In order that an ob- 
ject may be plainly visible to the naked eye, it must sub- 
tend an angle of nearly V. Consequently, a magnifying 
power of 60 is required to render a round object one mile 
in diameter on the surface of the moon plainly visible. 
Starting from this fact, we may readily form the follow- 
ing table, showing the diameters of the smallest objects 
that can be seen with different magnifying powers, always 
assuming that vision with these powers is perfect : 

Power 60 ; diameter of object 1 mile. 
Power 150 ; diameter 2000 feet. 
Power 500 ; diameter 600 feet. 
Power 1000 ; diameter 300 feet. 
Power 2000 ; diameter 150 feet. 

If telescopic power could be increased indefinitely, there 
would of course be no limit to the minuteness of an ob- 
ject visible on the moon's surface. But the necessary 
imperfections of all telescopes are such that only in extra- 
ordinary cases can any thing be gained by increasing the 



328 ASTRONOMY. 

magnifying power beyond 1000. The influence of warm 
and cold currents in our atmosphere is such as will for- 
ever prevent the advantageous use of high magnifying 
powers. After a certain limit we see nothing more by 
increasing the power, vision becoming indistinct in pro- 
portion as the power is increased. It may be doubted 
whether the moon was ever seen through a telescope to so 
good advantage as she would be seen with a magnifying 
power of 500, unaccompanied by any drawback from at- 
mospheric vibrations or imperfection of the telescope. 
In other words, it is hardly likely that an object less than 
600 feet in extent can ever be seen on the moon by any 
telescope whatever, unless it becomes possible to mount the 
instrument above the atmosphere of the earth. It is there- 
fore only the great features on the surface of the moon, 
and not the minute ones, which can be made out with the 
telescope. 

Character of the Moon's Surface. — The most striking 
point of difference between the earth and moon is seen in 
the total absence from the latter of any thing that looks 
like an undulating surface. JSIo formations similar to our 
valleys and mountain-chains have been detected. The 
lowest surface of the moon which can be seen with the 
telescope appears to be nearly smooth and flat, or, to 
speak more exactly, spherical (because the moon is a 
sphere). This surface has different shades of color in 
different regions. Some portions are of a bright, silvery 
tint, while others have a dark gray appearance. These dif- 
ferences of tint seem to arise from differences of material. 

Upon this surface as a foundation are built numerous 
formations of various sizes, but all of a very simple char- 
acter. Their general form can be made out by the aid of 
Fig. 89, and their dimensions by the scale of miles at 
the bottom of it. The largest and most prominent 
features are known as craters. They have a typical form 
consisting of a round or oval rugged wall rising from the 
plane in the manner of a circular fortification. These 



THE MOON'S SURFACE. 



329 



walls are frequently from three to six thousand metres in 
height, very rough and broken. In their interior we see 




Fig. 89. — aspect of the moon's surface. 

the plane surface of the moon already described. It is, 
however, generally covered with fragments or broken up 



330 ASTRONOMY. 

by small inequalities so as not to be easily made out. In 
the centre of the craters we frequently find a conical for- 
mation rising up to a considerable height, and much larger 
than the inequalities just described. In the craters we 
have a vague resemblance to volcanic formations upon the 
earth, the principal difference being that their magnitude 
is very much greater than any thing known here. The 
diameter of the larger ones ranges from 50 to 200 kilo- 
metres, while the smallest are so minute as to be hardly 
visible with the telescope. 

When the moon is only a few days old, the sun's rays 
strike very obliquely upon the lunar mountains, and they 
cast long shadows. From the known position of the sun, 
moon, and earth, and from the measured length of these 
shadows, the heights of the mountains can be calculated. 
It is thus found that some of the mountains near the south 
pole rise to a height of 8000 or 9000 metres (from 25,000 
to 30,000 feet) above the general surface of the moon. 
Heights of from 3000 to 7000 metres are very common 
over almost the whole lunar surface. 

Next to the so-called craters visible on the lunar disk, 
the most curious features are certain long bright streaks, 
which the Germans call rills or furrows. These extend 
in long radiations over certain of the craters, and have the 
appearance of cracks in the lunar surface which have been 
subsequently filled by a brilliant white material. Na- 
smyth and Carpenter have described some experiments 
designed to produce this appearance artificially. They 
took hollow glass globes, filled them with water, and heat- 
ed them until the surface was cracked. The cracks gen- 
erated at the weakest point of the surface radiate from the 
point in a manner strikingly similar in appearance to the 
rills on the moon. It would, however, be premature to 
conclude that the latter were actually produced in this 
way. 

The question of the origin of the lunar features has a 
bearing on theories of terrestrial geology as well as upon 



LIGHT AND HEAT OF THE MOON. 331 

various questions respecting the past history of the moon 
itself. It has been considered in this aspect by various 
geologists. 

Lunar Atmosphere. — The question whether the moon 
has an atmosphere has been much discussed. The only 
conclusion which has yet been reached is that no positive 
evidence of an atmosphere has ever been obtained, and 
that if one exists it is certainly several hundred times rarer 
than the atmosphere of our earth. The most delicate 
method of detecting one is to determine whether it 
will refract the light of a star seen through it. As the 
moon advances in her monthly course around the earth, she 
frequently appears to pass over bright stars. These phe- 
nomena are called occultations. Just before the limb of 
the moon appears to reach the star, the latter will be seen 
through the moon's atmosphere, if there is one, and will 
be displaced in a direction from the moon's centre. But 
the most careful observations have failed to show the 
slightest evidence of any such displacement. Hence the 
most delicate test for a lunar atmosphere gives no evi- 
dence whatever that it exists. 

The spectra of stars when about to be occulted have 
also been examined in order to see whether any absorption 
lines which might be produced by the lunar atmosphere 
became visible. The evidence in this direction has also 
been negative. Moreover, the spectrum of the moon itself 
does not seem to differ in the slightest from that of the 
sun. We conclude therefore that if there is a lunar at- 
mosphere, it is too rare to exert any sensible absorption 
upon the rays of light. 

Light and Heat of the Moon. — Many attempts have 
been made to measure the ratio of the light of the full 
moon and that of the sun. The results have been very 
discordant, but all have agreed in showing that the sun 
emits several hundred thousand times as much light as the 
full moon. The last and most careful determination is 



332 ASTRONOMY. 

that of Zollner, who finds the sun to be 618,000 times as 
bright as the full moon. 

The moon must reflect the heat as well as the light of 
the sun, and must also radiate a small amount of its own 
heat. But the quantities thus reflected and radiated are so 
minute that they have defied detection except with the 
most delicate instruments of research now known. By col- 
lecting the moon's rays in the focus of one of his large re- 
flecting telescopes, Lord Rosse was able to show that a 
certain amount of heat is actually received from the 
moon, and that this amount varies with the moon's phase, 
as it should do. He also sought to learn how much of 
the moon's heat was reflected and how much radiated. 
This he did by ascertaining its capacity for passing 
through glass. It is well known to students of physics 
that a very much larger portion of the heat radiated by 
the sun or other extremely hot bodies will pass through 
glass than of heat radiated by a cooler body. Experiments 
show that about 86 per cent of the sun's heat will pass 
through ordinary optical glass. If the heat of the moon 
were entirely reflected sun heat, it w^ould possess the same 
property, and the same proportion w T ould pass through 
glass. But the experiments of Lord Rosse have shown 
that instead of 86 percent, only 1 2 per cent passed through 
the glass. As a general result of all his researches, it may 
be supposed that about six sevenths of the heat given out 
by the moon is radiated and one seventh reflected. 

Is there any change on the surface of the Moon? — 
"When the surface of the moon was first found to be cov- 
ered by craters having the appearance of volcanoes at the 
surface of the earth, it was very naturally thought that 
these supposed volcanoes might be still in activity, and ex- 
hibit themselves to our telescopes by their flames. Sir 
William Herschel supposed that he saw several such vol- 
canoes, and, on his authority, they were long believed to 
exist. Subsequent observations have shown that this was 
a mistaken opinion, though a very natural one under the 



CHANGES ON THE MOON. 333 

circumstances. If we look at the moon with a telescope 
when she is three or four days old, we shall see the darker 
portion of her surface, which is not reached by the sun's 
rays, to be faintly illuminated by light reflected from the 
earth. This appearance may always be seen at the right 
time with the naked eye. If the telescope has an aperture 
of five inches or upward, and the magnifying power does 
not exceed ten to the inch, we shall generally see one or 
more spots on this dark hemisphere of the moon so much 
brighter than the rest of the surface that they may well 
suggest the idea of being self-luminous. It is, however, 
known that these are only spots possessing the power of 
reflecting back an unusually large portion of the earth's 
light. Not the slightest sound evidence of any incandes- 
cent eruption at the moon's surface has ever been found. 

Several instances of supposed changes on the moon's 
surface have been described in recent times. A few years 
ago a spot known as Linnaeus, near the centre of the 
moon's .visible disk, was found to present an appearance 
entirely different from its representation on the map of 
Beer and Maedler, made forty years before. More 
recently Klein, of Cologne, supposed himself to have dis- 
covered a yet more decided change in another feature of 
the moon's surface. 

The question whether these changes are proven is one 
on which the opinions of astronomers differ. The difficul- 
ty of reaching a certain conclusion arises from the fact that 
each feature necessarily varies in appearance, owing to the 
different ways in which the sun's light falls upon it. 
Sometimes the changes are very difficult to account for, 
even when it is certain that they do not arise from any 
change on the moon itself. Hence while some regard the 
apparent changes as real, others regard them as due only 
to differences in the mode of illumination. 



CHAPTER V. 

THE PLANET MAES. 
§ 1. DESCRIPTION OF THE PLANET. 

Mars is the next planet beyond the earth in the order 
of distance from the sun, being about half as far again as 
the earth. It has a decided red color, by which it may 
be readily distinguished from all the other planets. 
Owing to the considerable eccentricity of its orbit, its 
distance, both from the sun and from the earth, varies in a 
larger proportion than does that of the other outer planets. 

At the most favorable oppositions, its distance from the 
earth is about 0-38 of the astronomical unit, or, in round 
numbers, 57,000,000 kilometres (35,000,000 of miles). 
This is greater than the least distance of Venus, but we 
can nevertheless obtain a better view of Mars under these 
circumstances than of Venus, because when the latter is 
nearest to us its dark hemisphere is turned toward us, 
while in the case of Mars and of the outer planets the 
hemisphere turned toward us at opposition is fully illu- 
minated by the sun. 

The period of revolution of Mars around the sun is a 
little less than two years, or, more exactly, 687 days. The 
successive oppositions occur at intervals of two years and 
one or two months, the earth having made during this 
interval a little more than two revolutions around the sun, 
and the planet Mars a little more than one. The dates 
of several past and future oppositions are shown in the 
following table : 



OPPOSITIONS OF MARS. 335 

1871 March 20th. 

1873 April 27th. 

1875 June 20th. 

1877 September 5th. 

1879 , November 12th. 

1881 December 26th. 

1884 January 31st. 

1886 March 6th. 

Owing to the unequal motion of the planet, arising from 
the eccentricity of its orbit, the intervals between suc- 
cessive oppositions vary from two years and one month to 
two years and two and a half months. 

About August 26th of each year the earth is in the same 
direction from the sun as the perihelion of the orbit of 
Mars. Hence if an opposition occurs about that time, 
Mars will be very near its perihelion, and at the least 
possible distance from the earth. At the opposite season 
of the year, near the end of February, the earth is on 
the line drawn from the sun to the aphelion of the orbit 
Mars. The least favorable oppositions are therefore 
those which occur in February. The distance of Mars is 
then about 0-65 of the astronomical unit. 

The favorable oppositions occur at intervals of 15 or 
17 years, the period being that required for the successive 
increments of one or two months between the times of the 
year at which successive oppositions occur to make up an 
entire year. This will be readily seen from the preceding 
table of the times of opposition, which shows how the op- 
positions ranged through the entire year between 1871 
and 1886. The opposition of 1877 was remarkably fa- 
vorable. The next most favorable opposition will occur 
in 1892. 

Mars necessarily exhibits phases, but they are not so 
well marked as in the case of Venus, because the hemi- 
sphere which it presents to the observer on the earth is 
always more than half illuminated. The greatest phase 



336 ASTRONOMY. 

occurs when its direction is 90° from that of the sun, and 
even then six sevenths of its disk is illuminated, like that 
of the moon, three days before or after full moon. The 
phases of Mars were observed by Galileo in 1610, who, 
however, could not describe them with entire certainty. 

Rotation of Mars. — The early telescopic observers 
noticed that the disk of Mars did not appear uniform in 
color and brightness, but had a variegated aspect. In 
1666 the celebrated Dr. Kobert Hooke found that the 
markings on Mars were permanent and moved around in 
such a way as to show that the planet revolved on its axis. 
The markings given in his drawing can be traced at the 
present day, and are made use of to determine the exact 
period of rotation of the planet. Drawings made by 
Huyghens about the same time have been used in the 
same way. So well is the rotation fixed by them that the 
astronomer can now determine the exact number of times 
the planet has rotated on its axis since these old drawings 
were made. The period has been found by Mr. Proctor 
to be 2tt h 37 m 22 s -7, a result winch appears certain to one 
or two tenths of a second. It is therefore less than an 
hour greater than the period of rotation of the earth. 

Surface of Mars. — The most interesting result of these 
markings on Mars is the probability that its surface is di- 
versified by land and water, covered by an atmosphere, 
and altogether very similar to the surface of the earth. 
Some portions of the surface are of a decided red color, 
and thus give rise to the well-known fiery aspect of the 
planet. Other parts are of a greenish hue, and are there- 
fore supposed to be seas. The most striking features are 
two brilliant white regions, one lying around each pole of the 
planet. It has been supposed that this appearance is due 
to immense masses of snow and ice surrounding the poles. 
If this were so, it would indicate that the processes of evap- 
oration, cloud formation, and condensation of vapor into 
rain and snow go on at the surface of Mars as at the sur- 
face of the earth. A certain amount of color is given to 



ASPECT OF MARS. 337 

this theory by supposed changes in the magnitude of 
these ice-caps. But the problem of establishing such 
changes is one of extreme difficulty. The only way in 
which an adequate idea of this difficulty can be formed is 
by the reader himself looking at Mars through a telescope. 
If he will then note how hard it is to make out the 
different shades of light and darkness on the planet, and 



Fig. 90.— telescopic view op mars. 

how they must vary in aspect under -different conditions 
of clearness in our own atmosphere, he will readily per- 
ceive that much evidence is necessary to establish great 
changes. All we can say, therefore, is that the formation 
of the ice-caps in winter and their melting in summer has 
some evidence in its favor, but is not yet completely 
proven. 



338 ASTRONOMY. 

§ 2. SATELLITES OF MARS. 

Until the year 1877, Mars was supposed to have no sat- 
ellites, none having ever been seen in the most powerful 
telescopes. But in August of that year, Prof essor Hall, 
of the !N aval Observatory, instituted a systematic search 
with the great equatorial, which resulted in the discovery 
of two such objects. We have already described the op- 
position of 1877 as an extremely favorable one ; otherwise 
it would have been hardly possible to detect these bodies. 
They had never before been seen, partly on account of 
their extreme minuteness, which rendered them invisible 
except with powerful instruments and at the most favor- 
able times, and partly on account of the fact, already al- 
luded to, that the favorable oppositions occur only at inter- 
vals of 15 or 17 years. There are only a few weeks dur- 
ing each of these intervals when it is practicable to distin- 
guish them. 

These satellites are by far the smallest celestial bodies 
known. It is of course impossible to measure their diam- 
eters, as they appear in the telescope only as points of 
light. A very careful estimate of the amount of light 
which they reflect was made by Professor E. C. Picker- 
ing, Director of the Harvard College Observatory, who 
calculated how large they ought to be to reflect this light. 
He thus found that the outer satellite was probably about 
six miles and the inner one about seven miles in diameter, 
supposing them to reflect the solar rays precisely as Mars 
does. The outer one was seen with the telescope at a dis- 
tance from the earth of 7,000,000 times this diameter. 
The proportion would be that of a ball two inches in di- 
ameter viewed at a distance equal to that between the 
cities of Boston and JS"ew York. Such a feat of telescopic 
seeing is well fitted to give an idea of the power of modem 
optical instruments. 

Professor Hall found that the outer satellite, which 
he called Deimos, revolves around the planet in 30 h 16 m , 



SATELLITES OF MARS. 339 

and the inner one, called PJiobos, in 7 h 38 m . The latter is 
only 5800 miles from the centre of Mars, and less than 
4000 miles from its surface. It would therefore be almost 
possible with one of our telescopes on the surface of Mars 
to see an object the size of a large animal on the satellite. 
This short distance and rapid revolution make the inner 
satellite of Mars one of the most interesting bodies with 
which we are acquainted. It performs a revolution in its 
orbit in less than half the time that Mars revolves on its 
axis. In consequence, to the inhabitants of Mars, it 
would seem to rise in the west and set in the east. It will 
be remembered that the revolution of the moon around 
the earth and of the earth on its axis are both from w T est 
to east ; but the latter revolution being the more rapid, the 
apparent diurnal motion of the moon is from east to west. 
In the case of the inner satellite of Mars, however, this 
is reversed, and it therefore appeal's to move in the actual 
direction of its orbital motion. The rapidity of its phases 
is also equally remarkable. It is less than two hours from 
new moon to first quarter, and so on. Thus the inhabit- 
ants of Mars may see their inner moon pass through all 
its phases in a single night. 



CHAPTER VI. 

THE MINOR PLANETS. 

When the solar system was first mapped out in its true 
proportions by Copernicus and Kepler, only six primary 
planets were known — namely, Mercury, Venus, the 
Earth, Mars, Jupiter, and Saturn. These succeeded 
each other according to a nearly regular law, as we have 
shown in Chapter I., except that between Mars and Jupi- 
ter a gap was left, where an additional planet might be 
inserted, and the order of distance be thus made complete. 
It was therefore supposed by the astronomers of the seven- 
teenth and eighteenth centuries that a planet might be 
found in this region. A search for this object was insti- 
tuted toward the end of the last century, but before it 
had made much progress a planet in the place of the one 
so long expected was found by Piazzi, of Palermo. The 
discovery was made on the first day of the present century, 
1801, January 1st. 

In the course of the following seven years the astronom- 
ical world was surprised by the discovery of three other 
planets, all in the same region, though not revolving in 
the same orbits. Seeing four small planets where one 
large one ought to be, Olbers was led to his celebrated 
hypothesis that these bodies were the fragments of a large 
planet which had been broken to pieces by the action of 
some unknown force. 

A generation of astronomers now passed away without 
the discovery of more than these four. But in December, 
1845, PIknoke, of Dreisen, being engaged in mapping 



THE MINOR PLANETS. 341 

down the stars near the ecliptic, found a fifth planet of 
the group. In 1847 three more were discovered, and 
discoveries have since been made at a rate which thus far 
shows no signs of diminution. The number has now 
reached 200, and the discovery of additional ones seems to 
be going on as fast as ever. The frequent announcements 
of the discovery of planets which appear in the public 
prints all refer to bodies of this group. 

The minor planets are distinguished from the major 
ones by many characteristics. Among these we may 
mention their great number, which exceeds that of all the 
other known bodies of the solar system ; their small size ; 
their positions, all being situated between the orbits of 
Mars and Jupiter / the great eccentricities and inclina- 
tions of their orbits. 

Number of Small Planets. — It would be interesting to 
know how many of these planets there are in all, but it is 
as yet impossible even to guess at the number. As 
already stated, fully 200 are now known, and the number 
of new ones found every year ranges from 7 or 8 to 10 or 
12. If ten additional ones are found every year during 
the remainder of the century, 400 will then have been 
discovered. 

The discovery of these bodies is a very difficult work, 
requiring great practice and skill on the part of the as- 
tronomer. The difficulty is that of distinguishing them 
amongst the hundreds of thousands of telescopic stars 
which are scattered in the heavens. A minor planet 
presents no sensible disk, and therefore looks exactly like 
a small star. It can be detected only by its motion among 
the surrounding stars, which is so slow that hours or even 
days must elapse before it can be noticed. 

Magnitudes. — In consequence of the minor planets hav- 
ing no visible disks in the most powerful telescopes, it is im- 
possible to make any precise measurement of their diam- 
eters. These can, however, be estimated by the amount 
of light which the planet reflects. Supposing the propor- 



342 ASTRONOMY. 

tion of light reflected about the same as in the case of the 
larger planets, it is estimated that the diameters of the 
three or four largest, which are* those first discovered, 
range between 300 and 600 kilometres, while the smallest 
are probably from 20 to 50 kilometres in diameter. The 
average diameter of all that are known is perhaps less than 
150 kilometres — that is, scarcely more than one hundredth 
that of the earth. The volumes of solid bodies vary as the 
cubes of their diameters ; it might therefore take a million 
of these planets to make one of the size of the earth. 

Form of Orbits.— The orbits of the minor planets are much 
more eccentric than those of the larger ones ; their distance from 
the sun therefore varies very widely. The most eccentric orbit yet 
known is that of Aethra, which was discovered by Professor Wat- 
son in 1873. Its least distance from the sun is 161, a very little 
further than Mars, while at aphelion it is 3 '59, or more than twice 
as far. Two or three others are twice as far from the sun at aphe- 
lion as at perihelion, while nearly all are so eccentric that if the 
orbits were drawn to a scale, the eye would readily perceive that the 
sun was not in their centres. The largest inclination of all is that 
of Pallas, which is one of the original four, having been discovered 
by Olbers in 1802. The inclination to the ecliptic is 34°, or more 
than one third of a right angle. Five or six others have inclinations 
exceeding 20°; they therefore range entirely outside the zodiac, and 
in fact sometimes culminate to the north of our zenith. 

Origin of the Minor Planets.— The question of the origin of 
these bodies was long one of great interest. The features which we 
have described associate themselves very naturally with the cele- 
brated hypothesis of Olbers, that we here have the fragments of a 
single large planet which in the beginning revolved in its proper 
place between the orbits of Mars and Jupiter. Olbers himself sug- 
gested a test of his theory. If these bodies were really formed by 
an explosion of the large one, the separate orbits of the fragments 
would all pass through the point where the explosion occurred. A 
common point of iutersection was therefore long looked for ; but 
although two or three of the first four did pass pretty near each 
other, the required point could not be found for all four. 

It was then suggested that the secular changes in the orbits pro- 
duced by the action of the other planets would in time change the 
positions of all the orbits in such a way that they would no longer 
have any common intersection. The secular variations of their orbits 
were therefore computed, to see if there was any sign of the required 
intersection in past ages, but none could be found. No support 
has been given to Olbers' hypothesis by subsequent investigations, 
and it is no longer considered by astronomers to have any founda- 
tion. So far as can be judged, these bodies have been revolving 
around the sun as separate planets ever since the solar system itself 
was formed. 



CHAPTER VII. 

JUPITER AND HIS SATELLITES. 
§ 1. THE PLANET JUPITER. 

Jupiter is much the largest planet in the system. His 
mean distance is nearly 800,000,000 kilometres (480,000,- 
000 miles). His diameter is 140,000 kilometres, corre- 
sponding to a mean apparent diameter, as seen from the 
sun of 36" -5. His linear diameter is about y 1 ^, his surface 
is t -£-q , and his volume T oV o" tna ^ °^ tne sun - -^ s mass is 
1 \ 8 ,' and his density is thus nearly the same as the sun's — 
viz. ,0-24 of the earth's. He rotates on his axis in 9 h 55 m 20 s . 

He is attended by four satellites, which were discovered 
by Galileo on January 7th, 1610. He named them in 
honor of the Medicis, the Medicean stars. These satellites 
were independently discovered on January 16th, 1610, by 
Harriot, of England, who observed them through several 
subsequent years. Simon Marius also appears to have 
early observed them, and the honor of their discovery is 
claimed for him. They are now known as Satellites I, 
II, III, and IY, I being the nearest. 

The surface of Jupiter has been carefully studied with 
the telescope, particularly within the past 20 years. Al- 
though further from us than liars, the details of his disk 
are much easier to recognize. The most characteristic 
features are given in the drawings appended. These feat- 
ures are, firstly, the dark bands of the equatorial regions, 
and, secondly, the cloud-like forms spread over nearly the 
whole surface. At the limb all these details become indis- 



344 ASTRONOMY. 

tinct, and finally vanish, thus indicating a highly absorptive 
atmosphere. The light from the centre of the disk is twice 
as bright as that from the poles (Arago). The bands can 
be seen with instruments no more powerful than those 
used by Galileo, yet he makes no mention of them, al- 
though they were seen by Zucchi, Fontana, and others be= 
fore 1033. IIuyghens (1659) describes the bands as 
brighter than the rest of the disk — a unique observation, 
on which we must look with some distrust, as since 1660 
they have constantly been seen darker than the rest of the 
planet. 

The color of the bands is frequently described as a brick- 
red, but one of the authors has made careful studies in 




Fig. 91. — telescopic view of jupiter and his satellites. 

color of this planet, and finds the prevailing tint to be a 
salmon color, exactly similar to the color of Mars. The 
position of the bands varies in latitude, and the shapes of 
the limiting curves also change from day to day ; but in 
the main they remain as permanent features of the region 
to which they belong. Two such bands are usually vis- 
ible, but often more are seen. For example, C assist 
(1690, December 16th) saw six parallel bands extending 
completely around the planet. Herschel, in the year 
1793, attributed the aspects of the bands to zones of the 
planet's atmosphere more tranquil and less filled with 
clouds than the remaining portions, so as to permit the 



ASPECT OF JUPITER. 345 

true surface of the planet to be seen through these zones, 
while the prevailing clouds in the other regions give 
a brighter tint to these latter. The color of the bands 
seems to vary from time to time, and their bordering 
lines sometimes alter with such rapidity as to show that 
these borders are formed of something like clouds. 

The clouds themselves can easily be seen at times, and 
they have every variety of shape, sometimes appearing as 




Fig. 92. — telescopic view op jupiter, with a satellite and 
its SHADOW seen on it. 

brilliant circular white masses, but oftenerthey are similar 
in form to a series of white cumulous clouds such as are 
frequently seen piled up near the horizon on a summer's 
day. The bands themselves seem frequently to be veiled 
over with something like the thin cirrus clouds of our 
atmosphere. On one occasion an annulus of white cloud 
was seen on one of the dark bands for many days, retain- 
ing its shape through the whole period. 



346 ASTRONOMY. 

Such clouds can be tolerably accurately observed, and 
may be used to determine the rotation time of the planet. 
These observations show that the clouds have often a 
motion of their own, which is also evident from other con- 
siderations. 

The following results of observation of spots situated in 
various regions of the planet will illustrate this : 

h. m. 8. 
Cassini 1665, rotation time = 9 56 00 



Herschel 1778, 

Herschel 1779, 

schroeter 1785, 

Beer & Madler 1835, 

Airy 1835, 

Schmidt 1863, 



= 9 55 40 

= 9 50 48 

= 9 56 56 

= 9 55 26 

=9 55 21 

= 9 55 29 



§ 2. THE SATELLITES OF JUPITER. 

Motions of the Satellites. — The four satellites move 
about Jupiter from west to east in nearly circular orbits. 
When one of these satellites passes between the sun and 
Jupiter, it casts a shadow upon Jupiter 's disk (see Fig. 92) 
precisely as the shadow of our moon is thrown upon the 
earth in a solar eclipse. If the satellite passes through 
Jupiter f s own shadow in its revolution, an eclipse of this 
satellite takes place. If it passes between the earth and 
Jupiter, it is projected upon Jupiter's disk, and we ha ye a 
transit ; if Jupiter is between the earth and the satellite, 
an occultation of the latter occurs. All these phenomena 
can be seen from the earth with a common telescope, and 
the times of observation are all found predicted in the 
Nautical Almanac. In this way we are sure that the black 
spots which we see moving across the disk of Jupiter are 
really the shadows of the satellites themselves, and not phe- 
nomena to be otherwise explained. These shadows being 
seen black upon Jupiter's surface, show that this planet 
shines by reflecting the light of the sun. 



SATELLITES OF JUPITER. 347 

Telescopic Appearance of the Satellites. — Under ordi- 
nary circumstances, the satellites of Jupiter are seen to 
have disks — that is, not to be mere points of light. Un- 
der very favorable conditions, markings have beeen seen 
on these disks, and it is very curious that the anomalous 
appearances given in Fig. 93 (by Dr. Hastings; have been 
seen at various times by other good observers, as Secchi, 
Dawes, and Rutherfurd. Satellite III, which is much 
the largest, has decided markings on its face ; IV some- 
times appears, as in the figure, to have its circular outline 



%■ • • 



m w i 

Fig. 93. — telescopic appearance of jupitek's satellites. 

cut off by right lines, and satellite I sometimes appears 
gibbous. The opportunities for observing these appear- 
ances are so rare that nothing is known beyond the bare 
fact of their existence, and no plausible explanation of the 
figure shown in IV has been given. 

Phenomena of the Satellites. — The phenomena of the satel- 
lites are illustrated in Fig. 94. Here S represents the sun, A T 
the orbit of the earth (the earth itself being at T), the outer circle 
the orbit of Jupiter, and the four small circles upon the latter four 
different positions of the orbit of a satellite. In the centre of each 
of the satellite orbits will be seen a small white circle designed to 
represent the planet Jupiter itself. The dotted lines drawn from 
each edge of the sun to the corresponding edges of the planet and 
continued until they meet in a point show the outlines of the 
shadow of Jupiter. 

Let us first consider the position of Jupiter marked J to the left 
of the figure, it being then in opposition to the sun. The observer 
on the earth at T could not then see an object anywhere in the 
shadow of Jupiter because the latter is entirely behind the planet. 
Hence, as the satellite moves around, he will see it disappear behind 
the right-hand limb of the planet and reappear from the left-hand 
limb. Such a phenomenon is called an oecultation, and is desig- 
nated as disappearance or reappearance, according to the phase. 

It may be remarked, however, that the inclination of the outer 
satellite to the orbit of Jupiter is so great that it sometimes passes 



348 



ASTRONOMY. 



entirely above or below the planet, and therefore is not occulted 
at all. 

Let us next consider Jupiter in the position J'' near the bottom of 
the figure, the shadow, as before, pointing from the planet directly 
away from the sun. If the shadow were a visible object, the ob- 
server on the earth at T could see it projected out on the right of 
the planet, because he is not in the line between Jupiter and the sun. 
Hence as a satellite moves around and enters the shadow, he will see 
it disappear from sight, owing to the sunlight being cut off ; this 




Tig. 94.— phenomena of jupiter's satellites. 

is called an eclipse dimpvearnvce. If the satellite is one of the two 
outer ones, he will be able to see it reappear again after it comes 
out of the shadow before it is occulted behind the planet. 

Soon afterward the occultation will occur, and it will afterward 
reappear on the left. In the case of the inner or first satellite, how- 
ever, the point of emergence from the shadow is hidden behind the 
planet, consequently the observer, after it once disappears in the shad- 
ow, will not see it reappear until it emerges from behind the planet. 

If the planet is in the position J\ the satellite will be occulted 



SATELLITES OF JUPITER. 349 

behind the planet where it reaches the first dotted line. If it is the in- 
ner satellite, it will not be seen to reappear on the other side of the 
planet, because when it reaches the second dotted line it has entered 
the shadow. After a while, however, it will reappear from the 
shadow some little distance to the left of the planet ; this phe- 
nomenon is called an eclipse reappearance. In the case of the outer 
satellites, it may sometimes happen that they are visible for a short 
time after they emerge from behind the disk and before they enter 
the shadow. 

These different appearances are, for convenience, represented in 
the figure as corresponding to different positions of Jupiter in his 
orbit, the earth having the same position in all ; but since JvpiUr 
revolves around the sun only once in twelve years, the changes of 
relative position really correspond to different positions of the earth 
in its orbit during the course of the year. 

The satellites completely disappear from telescopic view when 
they enter the shadow of the planet. This seems to show that 
neither planet nor satellite is self-luminous to any great extent. If the 
satellite were self-luminous, it would be seen by its own light, and 
if the planet were luminous the satellite might be seen by the re- 
flected light of the planet. 

The motions of these objects are connected by two curious and 
important relations discovered by La Place, and expressed as fol- 
lows: 

I. The mean motion of the first satellite added to twice the mean 
motion xf the third is exactly equal to three times the mean motion of 
the second. 

II. If to the mean longitude of the first satellite we add twice the 
mean longitude of the third, and subtract three times the mean longitude 
of the second, the difference is always 180°. 

The first of these relations is shown in the following table of the 
mean daily motions of the satellites: 

Satellite I in one day moves 203°-4890 

II " «' lOr-3748 

" III " " 50-3177 

" IV " " 21 c -5711 

Mo ion of Satellite 1 203°-4890 

Twite that of Satellite III 100° -6354 

Sum 304° • 1 244 

Three times motion of Satellite II 304°«.P244 

Observations showed that this condition was fulfilled as exactly 
as possible, but the discovery of La Place consisted in showing that 
if the approximate coincidence of the mean motions was once es- 
tablished, they could never deviate from exact coincidence with 
the law. The case is analogous to that of the moon, which always 
presents the same face to us and which always will since the rela- 
tion being once ajmroximately true, it will become exact and ever 
remain so. 



350 



ASTRONOMY. 



The discovery of the gradual propagation of light by means of 
these satellites has already been described, and it has also been ex- 
plained that they are of use in the rough determination of longi- 
tudes. To facilitate their observation, the Nautical Almanac gives 
complete ephemerides of their phenomena. A specimen of a por- 
tion of such an ephemeris for 1865, March 7th, 8th, and 9th, is 
added. The times are Washington mean times. Tli6 letter W in- 
dicates that the phenomenon is visible in Washington. 

1865— Makch. 











d. h. 


7/1. S 


I. 


Eclipse 


Disapp. 




7 18 


27 38-5 


I. 


Occult. 


Reapp. 




7 21 


56 


III. 


Shadow 


Ingress 




8 7 


27 


III. 


Shadow 


Egre>s 




8 9 


58 


III. 


Transit 


Ingress 




8 12 


31 


II. 


Eclipse 


Disapp. 




8 13 


1 22-7 


III. 


Transit 


Egress 


w. 


8 15 


6 


II. 


Eclipse 


Keapp. 


W. 


8 15 


24 11-1 


II. 


Occult. 


Disapp. 


w. 


8 15 


27 


I. 


Shadow 


Ingress 


w. 


8 15 


43 


I. 


Transit 


Ingress 


w. 


8 16 


58 


I. 


Shadow 


Egress 




8 17 


57 


II. 


Occult. 


Keapp. 




8 17 


59 


I. 


Transit 


Egress 




8 19 


13 


I. 


Eclipse 


Disapp. 




9 12 


55 59-4 


I. 


Occult. 


Keapp. 


w. 


9 16 


25 



Suppose an observer near New York City to have determined his 
local time accurately. This is about 13 m faster than Washington 
time. On 1865, March 8th, he would look for the reappearance of 
II at about 15 h 34 m of his local time. Suppose he observed it 
at 15 h 36 m 22 s -7 of his time : then his meridian is 12 m ll s -6 
east of Washington. The difficulty of observing these eclipses with 
accuracy, and the fact that the aperture of the telescope employed 
has an important effect on the appearances seen, have kept this 
method from a wide utility, which it at first seemed to promise. 

The apparent diameters of these satellites have been measured by 
Stkuve, Secchi, and others, and the best results are : 

I, r-0; II, 0"-9; III, l"-5; IV, l"-3. 

Their masses (Jupiter^t) are : 

I, 0000017 ; II, 000023 ; III, 000088 ; IV, 0*000043. 

The third satellite is thus the largest, and it has about the den- 
sity of the planet. The true diameters vary from 2200 to 3700 
miles. The volume of II is about that of our moon ; III approaches 
our earth in size. 

Variations in the light of these bodies have constantly been 
noticed which have been supposed to be due to the fact that they 
turned on their axes once in a revolution, and thus presented various 
faces to us. The recent accurate photometric measures of Engel- 
mam show that this hypothesis will not account for all the changes 
observed, some of which appear to be quite sudden. 



SATELLITES OF JUPITER. 



351 



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CHAPTER VIII. 

SATURN AND ITS SYSTEM 
§ 1. GENERAL DESCRIPTION. 

Saturn is the most distant of the major planets known 
to the ancients. It revolves around the sun in 29J years, 
at a mean distance of about 1,400,000,000 kilometres 
(882,000,000 miles). The angular diameter of the ball of 
the planet is about 16" • 2, corresponding to a true diam- 
eter of about 110,000 kilometres (70,500 miles). Its diam- 
eter is therefore nearly nine times and its volume about 
700 times that of the earth. It is remarkable for its small 
density, which, so far as known, is less than that of any 
other heavenly body, and even less than that of water. 
Consequently, it cannot be composed of rocks, like those 
which form our earth. It revolves on its axis, according 
to the recent observations of Professor Hall, in 10 h 14 nL 
21 s , or less than half a day. 

Saturn is perhaps the most remarkable planet in the so- 
lar system, being itself the centre of a system of its own, 
altogether unlike any thing else in the heavens. Its most 
noteworthy feature is seen in a pair of rings which sur- 
round it at a considerable distance from the planet itself. 
Outside of these rings revolve no less than eight satellites, 
or twice the greatest number known to surround any 
other planet. The planet, rings, and satellites are alto- 
gether called the Saturnian system. The general appear- 
ance of this system, as seen in a small telescope, is shown 
in Fig. 95. 



ASPECT OF SATURN. 353 

To the naked eye, Saturn is of a dull yellowish color, 
shining with about the brilliancy of a star of the first mag- 
nitude. It varies in brightness, however, with the way 
in which its ring is seen, being brighter the wider the ring 
appears. It comes into opposition at intervals of one year 
and from twelve to fourteen days. The following are the 
times of some of these oppositions, by studying which one 
will be enabled to recognize the planet : 



Fig. 95. — telescopic view op the saturnian system. 

1879 October 5th. 

1880 October 18th. 

1881 October 31st. 

1882 November 11th. 

1883 November 28th. 

1884 December 11th. 

During these years it will be best seen in the autumn 
and winter. 



354 ASTRONOMY. 

When viewed with a telescope, the physical appearance 
of the ball of Saturn is quite similar to that of Jupiter, 
having light and dark belts parallel to the direction of its 
rotation. But these cloud-like belts are very difficult to 
see, and so indistinct that it is not easy to determine the 
time of rotation from them. This has been done by ob- 
serving the revolution of bright or dark spots which appear 
on the planet on very rare occasions. 

§ 2. THE RINGS OP SATURN. 

The rings are the most remarkable and characteristic 
feature of the Saturnian system. Fig. 96 gives two views 
of the ball and rings. The upper one shows one of their 
aspects as actually presented in the telescope, and the 
lower one shows what the appearance would be if the 
planet were viewed from a direction at right angles to the 
plane of the ring (which it never can be from the earth). 

The first telescopic observers of /Saturn were unable to 
see the rings in their true form, and were greatly per- 
plexed to account for the appearance which the planet 
presented. Galileo described the planet as " tri-corpo- 
rate," the two ends of the ring having, in his imperfect 
telescope, the appearance of a pair of small planets at- 
tached to the central one. " On each side of old Saturn 
were servitors who aided him on his way." This sup- 
posed discovery was announced to his friend Kepler in 
the following logogriph : 

smaismrmilmepoetalevmibunenugttaviras, which, being 
transposed, becomes — 

" Altissimum planetam tergeminum observavi" (I have 
observed the most distant planet to be triform). 

The phenomenon constantly remained a mystery to its 
first observer. In 1610 he had seen the planet accompa- 
nied, as he supposed, by two lateral stars ; in 1612 the 
latter had vanished, and the central body alone remained. 
After that Galileo ceased to observe Saturn. 



RINGS OF SATURN. 



355 




Fig. 96.— ttinqs of satuhn. 



356 ASTRONOMY. 

The appearances of the ring were also incomprehensible 
to Hevelius, Gassendi, and others. It was not nntil 
1655 (after seven years of observation) that the celebrated 
Huyghens discovered the true explanation of the remark- 
able and recurring series of phenomena presented by the tri- 
corporate planet. 

He announced his. conclusions in the following logo- 
griph :— 

" aaaaaa ccccc d eeeee g h iiiiiii 1111 mm nnnnnnnnn 
oooo pp q rr s ttttt uuuuu, ' ' which, when arranged, read — 

" Annulo cingitur, tenui, piano, nusquam coherente, 
ad eclipticam inclinato" (it is girdled by a thin plane ring, 
nowhere touching, inclined to the ecliptic). 

This description is complete and accurate. 

In 1665 it was found by Ball, of England, that what 
Huyghens had seen as a single ring was really two. A 
division extended all the way around near the outer edge. 
This division is shown in the figures. 

In 1850 the Messrs. Bond, of Cambridge, found that there 
was a third ring, of a dusky and nebulous aspect, inside 
the other two, or rather attached to the inner edge of the 
inner ring.. It is therefore known as BondSs dusky ring. 
It had not been before fully described owing to its dark- 
ness of color, which made it a difficult object to see except 
with a good telescope. It is not separated from the bright 
ring, but seems as if attached to it. The latter shades off 
toward its inner edge, which merges gradually into the 
dusky ring so as to make it difficult to decide precisely 
where it ends and the dusky ring begins. The latter ex- 
tends about one half way from the inner edge of the 
bright ring to the ball of the planet. 

Aspect of the Rings. — As Saturn revolves around the 
sun, the plane of the rings remains parallel to itself. That 
is, if we consider a straight line passing through the centre 
of the planet, perpendicular to the plane of the ring, as 
the axis of the latter, this axis will always point in the 
same direction. In this respect, the motion is similar to 



BWGS OF SATURN. 



35? 



that of the earth around the sun. The ring of Saturn is 
inclined about 27° to the plane of its orbit. Conse- 
quently, as the planet revolves around the sun, there is a 
change in the direction in which the sun shines upon it 
similar to that which produces the change of seasons upon 
the earth, as shown in Fig. 46, page 109. 

The corresponding changes for Saturn are shown in 
Fig. 97. During each revolution of Saturn the plane 




Fig. 97. — different aspects of the ring of saturn as seek - 
from the earth. 

of the ring passes through the sun twice. This occurred 
in the years 1862 and 1878, at two opposite points of the 
orbit, as shown in the figure. At two other points, mid- 
way between these, the sun shines upon the plane of the 
ring at its greatest inclination, about 27°. Since the earth 
is little more than one tenth as far from the sun as Sat- 
urn is, an observer always sees Saturn nearly, but not 
quite, as if he were upon the sun. Hence at certain times 



358 ASTRONOMY. 

the rings of Saturn are seen edgeways, while at other 
times they are at an inclination of 27°, the aspect depend- 
ing npon the position of the planet in its orbit. The fol- 
lowing are the times of some of the phases : 

1878, February 7th.— The edge of the ring was turned 
toward the sun. It could then be seen only as a thin 
line of light. 

1885. — The planet having moved forward 90°, the south 
side of the rings may be seen at an inclination of 27°. 

1891, December. — The planet having moved 90° fur- 
ther, the edge of the ring is again turned toward the sun. 

1899. — The north side of the ring is inclined toward the 
sun, and is seen at its greatest inclination. 

The rings are extremely thin in proportion to their ex- 
tent. Rings cut out of a large newspaper would have much 
the same proportions as those of Saturn. Consequently, 
when their edges are turned toward the earth, they appear 
as a thin line of light, which can be seen only with power- 
ful telescopes. With such telescopes, the planet appears 
as if it were pierced through by a piece of very fine wire, 
the ends of which project on each side more than the diam- 
eter of the planet. It has frequently been remarked that 
this appearance is seen on one side of the planet, when no 
trace of the ring can be seen on the other. 

There is sometimes a period of a few weeks during 
which the plane of the ring, extended outward, passes be- 
tween the sun and the earth. That is, the sun shines on 
one side of the ring, while the other or dark side is turned 
toward the earth. In this case, it seems to be established 
that only the edge of the ring is visible. If this be so, 
the substance of the rings cannot be transparent to the 
sun's rays, else it would be seen by the light which passes 
through it. 

Possible Changes in the Rings.— In 1851 Otto Struve pro- 
pounded a noteworthy theory of changes going on in the rings of 
Saturn. From all the descriptions, figures, and measures given by 
the older astronomers, it appeared that two hundred years ago the 



RINGS OF SATURN. 359 

space between the planet and the inner ring was at least equal to 
the combined breadth of the two rings. At present this distance 
is less than one half of this breadth. Hence Struve concluded that 
the inner ring was widening on the inside, so that its edge had been 
approaching the planet at the rate of about 1"*3 in a century. The 
space between the planet and the inner edge of the bright ring is 
now about 4% so that if Struve 1 s theory were true, the inner edge 
of the ring would actually reach the planet about the year 2200. 
Notwithstanding the amount of evidence which Struve cited in 
favor of his theory, astronomers generally are incredulous respecting 
the reality of so extraordinary a change. The measures necessary 
to settle the question are so difficult and the change is so slow that 
some time must elapse before the theory can be established, even if 
it is true. The measures of Kaiser render this doubtful. 

Shadow of Planet and Ring.— With any good telescope it is 
easy to observe both the shadow of the ring upon the ball of Saturn 
and that of the ball upon the ring. The form which the shadows 
present often appears different from that which the shadow ought 
to have according to the geometrical conditions. These differences 
probably arise from irradiation and other optical illusions. 

Constitution of the Rings of Saturn.— The nature of these 
objects has been a subject both of wonder and of investigation by 
mathematicians and astronomers ever since they were discovered. 
They were at first supposed to be solid bodies ; indeed, from their 
appearance it was difficult to conceive of them as anything else. 
The question then arose : What keeps them from falling on the 
planet ? ' It was shown by La Place that a homogeneous and solid 
ring surrounding the planet could not remain in a state of equili- 
brium, but must be precipitated upon the central ball by the small- 
est disturbing force. Herschel having thought that he saw cer- 
tain irregularities in the figure of the ring, La Place concluded that 
the object could be kept in equilibrium by them. He simply as- 
sumed this, but did not attempt to prove it. 

About 1850 the investigation was again begun by Professors Bond 
and Peirce, of Cambridge. The former supposed that the rings 
could not be solid at all, because they had sometimes shown signs of 
being temporarily broken up into a large number of concentric 
rings. Although this was probably an optical illusion, he concluded 
that the rings must be liquid. Professor Peirce took up the prob- 
lem where La Place had left it, and showed that even an irregular 
solid ring would not be in equilibrium about Saturn. He therefore 
adopted the view of Bond, that the rings were fluid ; but finding 
that even a fluid ring would be unstable without a support, he sup- 
posed that such a support might be furnished by the satellites. 
This view has also been abandoned. 

It is now established beyond reasonable doubt that the rings do 
not form a continuous mass, but are really a countless multitude of 
small separate particles, each of which revolves on its own account. 
These satellites are individually far too small to be seen in any tele- 
scope, but so numerous that when viewed from the distance of the 
earth they appear as a continuous mass, like particles of dust float- 



360 ASTRONOMY. 

ing in a sunbeam. This theory was first propounded by Cassini, 
of Paris, in 1715. It had been forgotten for a century or more, 
when it was revived by Professor Clerk Maxwell in 1856. The 
latter published a profound mathematical discussion of the whole 
question, in which he shows that this hypothesis and this alone 
would account for the appearances presented by the rings. 

Kaiser's measures of the dimensions of the Saturnian system are : 

BALL OF SATURN. 

Equatorial diameter 17 '"274 

Polar " 15-"392 

rings. 

Major axis of outer ring 39* "471 

11 " " the great division 34* '227 

" " " the inner edge of ring 27'"859 

Width of the ring 5-806 

Dark space between ball and ring 5*"292 



§ 3. SATELLITES OF SATURN. 

Outside the rings of Saturn revolve its eight satellites, 
the order and discovery of which are shown in the following 
table : 



No. 


Name. 


Distance 
from 

Planet. 


Discoverer. 


Date of Discovery. 


1 
2 
3 
4 
5 
6 
7 
8 


Mimas. 

Enceladus. 

Tethys. 

Dione. 

Rhea. 

Titan. 

Hyperion. 

Japetus. 


3-3 
4-3 
5-3 

6-8 

9-5 

20-7 

26-8 

64-4 


Herschel. 

Herschel. 

Cassini. 

Cassini. 

Cassini. 

Huygheus. 

Bond. 

Cassini. 


1789, September 17. 
1789, August 28. 
1684, March. 
1684, March. 
1672, December 23. 
1655, March 25. 
1848, September 16. 
1671, October. 



The distances from the planet are given in radii of the 
latter. The satellites Mimas and Hyperion are visible 
only in the most powerful telescopes. The brightest of 
all is Titan, which can be seen in a telescope of the small- 
est ordinary size. Japetus has the remarkable peculiarity 



SATELLITES OF SATURN. 



361 



of appearing nearly as bright as Titan when seen west of 
the planet, and so faint as to be visible only in large tel- 
escopes when on the other side. This appearance is ex- 
plained by supposing that, like our moon, it always pre- 
sents the same face to the planet, and that one side of it is 
black and the other side white. When west of the planet, 
the bright side is turned toward the earth and the satellite is 
visible. On the other side of the planet, the dark side is 
turned toward us, and it is nearly invisible. Most of the 
remaining five satellites can be ordinarily seen with tele- 
scopes of moderate power. 

The elements of all the satellites are shown in the fol- 
lowing table : 



Satellite. 


Mean Daily 
Motion. 


Mean 
Distance 

from 
Saturn. 


Longitude 

of 
Peri-Sat. 


Eccen- 
tricity. 


Inclina- 
tion to 
Ecliptic. 


Longitude 

of 

Node 


Mimas 

Enceladus. 

Tetbys 

Diane 

Hhea 

Titan 

Hyperion. . 
Japetus. . . 


381-953 
262-721 
190-69773 
131-534930 
79-690216 
22-577033 
16-914 
4-538036 


'42*70 
54-60 
76-12 
176-75 
214-22 
514-64 


/ 

1 

% 

1 

1 

1 

257.16 

40-00 

351-25 


o / 

1 
1 

? 

1 

1 

•0286 
-125 

•0282 


o / 

28 00 
28 00 
28 10 
28 10 
28 11 

27 34 

28 00 
18 44 


/ 

168 00 
168 00 
167 38 
167 38 

166 34 

167 56 

168 00 
142 53 



CHAPTER IX. 

THE PLANET UKANUS. 

Uranus was discovered on March 13th, 1781, by Sir 
William Herschel (then an amateur observer) with a 
ten-foot reflector made by himself. He was examining a 
portion of the sky near H Geminorum, when one of the 
stars in the field of view attracted his notice by its pecu- 
liar appearance. On further scrutiny, it proved to have a 
planetary disk, and a motion of over 2" per hour. Her- 
schel at first supposed it to be a comet in a distant part 
of its orbit, and nnder this impression parabolic orbits 
were computed for it by various mathematicians. None 
of these, however, satisfied subsequent observation^ 
and it was finally announced by Lexell and La Place 
that the new body was a planet revolving in a nearly 
circular orbit. We can scarcely comprehend now the 
enthusiasm with which this discovery was received. No 
new body (save comets) had been added to the solar system 
since the discovery of the third satellite of Saturn in 1684, 
and all the major planets of the heavens had been known 
for thousands of years. 

Herschel suggested, as a name for the planet, Geor- 
gium Sidus, and even after 1800 it was known in the Eng- 
lish Nautical Almanac as the Georgian Planet. Lalande 
suggested Herschel as its designation, but this was judged 
too personal, and finally the name Uranus was adopted. 
Its symbol was for a time written Jj-I, in recognition of the 
name proposed by Lalande. 

Uranus revolves about the sun in 84 years. Its appar- 
ent diameter as seen from the earth varies little, being 



THE PLANET URANUS. 363 

about 3'« 9. Its true diameter is about 50,000 kilometres, 
and its figure is, so far as we yet know, exactly spherical. 

In physical appearance it is a small greenish disk with- 
out markings. It is possible that the centre of the disk is 
slightly brighter than the edges. At its nearest approach 
to the earth, it shines as a star of the sixth magnitude, 
and is just visible to an acute eye when the attention is 
directed to its place. In small telescopes with low pow- 
ers, its appearance is not markedly different from that of 
stars of about its own brilliancy. 

It is customary to speak of Herschel's discovery of 
Uranus as an accident ; but this is not entirely just, as 
all conditions for the detection of such an object, if it ex- 
isted, were fulfilled. At the same time the early identifi- 
cation of it as a planet was more easy than it would have 
been eleven days earlier, when, as Arago points out, the 
planet was stationary. 

Sir "William Herschel suspected that Uranus was ac- 
companied by six satellites. 

Of the ' existence of two of these satellites there has 
never been any doubt, as they were steadily observed by 
Heeschel from 1787 until 1810, and by Sir John Her- 
schel during the years 1828 to 1832, as well as by other 
later observers. None of the other four satellites de- 
scribed by Herschel have ever been seen by other ob- 
servers, and he was undoubtedly mistaken in supposing 
them to exist. Two additional ones were discovered by 
Lassell in 1847, and are, with the satellites of Mars, the 
faintest objects in the solar system. Neither of them is 
identical with any of the missing ones of Herschel. As 
Sir William Herschel had suspected six satellites, the 
following names for the true satellites are generally adopt- 
ed to avoid confusion : 

DATS. 

I, Ariel Period = 2-520383 

II, Umbriel " = 4.114181 

III, Titania, Herschel's (II.) " = 8-705897 

IV, Oberon, Herschel's (IV.) " = 13.4(>32t>9 



364: ASTRONOMY. 

It is an interesting question whether the observations 
which Herschel assigned to his supposititious satellite I 
may not be composed of observations sometimes of Ariel, 
sometimes of Umbriel. In fact, out of nine supposed 
observations of I, one case alone was noted by Herschel 
in which his positions were entirely trustworthy, and on 
this night Umbriel was in the position of his supposed 
satellite I. 

It is likely that Ariel varies in brightness on different 
sides of the planet, and the same phenomenon has also 
been suspected for Titania. 

The most remarkable feature of the satellites of Uranus is that 
their orbits are nearly perpendicular to the ecliptic instead of 
having a small inclination to that plane, like those of all the orbits 
of both planets and satellites previously known. To form a correct 
idea of the position of the orbits, we must imagine them tipped over 
until their north pole is nearly 8 8 below the ecliptic, instead of 90° 
above it. The pole of the orbit which should be considered as the 
north one is that from which, if an observer look down upon a re- 
volving body, the latter would seem to turn in a direction opposite 
that of the hands of a watch. When the orbit is tipped over more 
than a right angle, the motion from a point in the direction of the 
north pole of the ecliptic will seem to be the reverse of this ; it is 
therefore sometimes considered to be retrograde. This term is fre- 
quently applied to the motion of the satellites of Uranus, but is 
rather misleading, since the motion, being nearly perpendicular to 
the ecliptic, is not exactly expressed by the term. 

The four satellites move in the same plane, so far as the most re- 
fined observations have ever shown. This fact renders it highly 
probable that the planet Uranus revolves on its axis in the same 
plane with the orbits of the satellites, and is therefore an oblate 
spheroid like the earth. This conclusion is founded on the consid- 
eration that if the planes of the satellites were not kept together by 
some cause, they would gradually deviate from each other owing to 
the attractive force of the sun upon the planet. The different satel- 
lites would deviate by different amounts, and it would be extremely 
improbable that all the orbits would at any time be found in the 
same plane. Since we see them in the same plane, we conclude that 
some force keeps them there, and the oblateness of the planet would 
cause such a force. 



CHAPTER X. 

THE PLANET NEPTUNE. 

After the planet Uranus had been observed for some 
thirty years, tables of its motion were prepared by 
Bouvard. He had as data available for this purpose not 
only the observations since 1781, but also observations 
made by Le Monnier, Flamsteed, and others, extending 
back as far as 1695, in which the planet was observed for 
a fixed star and so recorded in their books. As one of 
the chief difficulties in the way of obtaining a theory of 
the planet's motion was the short period of time during 
which it had been regularly observed, it was to be sup- 
posed that these ancient observations would materially aid 
in obtaining exact accordance between the theory and ob- 
servation. But it was found that, after allowing for all 
perturbations produced by the known planets, the ancient 
and modern observations, though undoubtedly referring to 
the same object, were yet not to be reconciled with each 
other, but differed systematically. Bouvard was forced 
to omit the older observations in his tables, which were 
published in 1820, and to found his theory upon the 
modern observations alone. By so doing, he obtained a 
good agreement between theory and the observations of 
the few years immediately succeeding 1820. 

Bouvard seems to have formulated the idea that a possi- 
ble cause for the discrepancies noted might be the exist- 
ence of an unknown planet, but the meagre data at his 
disposal forced him to leave the subject untouched. In 
1830 it was found that the tables which represented the 



366 ASTRONOMY. 

motion of the planet well in 1820-25 were 20" in error, in 
1840 the error was 90", and in 1845 it was over 120". 

These progressive and systematic changes attracted the 
attention of astronomers to the subject of the theory of 
the motion of Uranus. The actual discrepancy (120") in 
1845 was not a quantity large in itself. Two stars of the 
magnitude of U'anus, and separated by only 120", would 
be seen as one to the unaided eye. It was on account of 
its systematic and progressive increase that suspicion was 
excited. Several astronomers attacked the problem in vari- 
ous ways. The elder Struve, at Pulkova, prosecuted a 
search for a new planet along with his double star obser- 
vations ; Bessel, at Koenigsberg, set a student of his own, 
Fleming, at a new comparison of observation with theo- 
ry, in order to furnish data for a new determination ; 
Arago, then Director of the Observatory at Paris, sug- 
gested this subject in 1845 as an interesting field of re- 
search to Le Yerrier, then a rising mathematician 
and astronomer. Mr. J. C. Adams, a student in Cam- 
bridge University, England, had become aware of the 
problems presented by the anomalies in the motion of 
Uranus, and had attacked this question as early as 1843. 
In October, 1845, Adams communicated to the Astrono- 
mer Royal of England elements of a new planet so situated 
as to produce the perturbations of the motion of Uranus 
which had actually been observed. Such a prediction 
from an entirely unknown student, as Adams then was, 
did not carry entire conviction with it. A series of acci- 
dents prevented the unknown planet being looked for by 
one of the largest telescopes in England, and so the mat- 
ter apparently dropped. It may be noted, however, that 
we now know Adams' elements of the new planet to have 
been so near the truth that if it had been really looked for 
by the powerful telescope which afterward discovered its 
satellite, it could scarcely have failed of detection. 

Bessel' s pupil Fleming died before his work was done, 
and Bessel's researches were temporarily brought to 



DISCO VEB T OF NEPTUNE. 367 

an end. Struve's search was unsuccessful. Only Le 
Y eerier continued his investigations, and in the most 
thorough manner. He first computed anew the pertur- 
bations of Uranus produced by the action of Jupiter and 
Saturn. Then lie examined the nature of the irregulari- 
ties observed. These showed that if they were caused by 
an unknown planet, it could not be between Saturn and 
Uranus, or else Saturn would have been more affected 
than was the case. 

The new planet was outside of Uranus if it existed at 
all, and as a rough guide Bode's law was invoked, which 
indicated a distance about twice that of Uranus. In the 
summer of 1846, Le Terrier obtained complete elements 
of a new planet, which would account for the observed 
irregularities in the motion of Uranus, and these were 
published in France. They were very similar to those of 
Adams, which had been communicated to Professor Chal- 
lis, the Director of the Observatory of Cambridge. 

A search was immediately begun by Challis for such 
an object, -and as no star-maps were at hand for this region 
of the sky, he began mapping the surrounding stars. In 
so doing the new planet was actually observed, both on 
August 4th and 12th, 1846, but the observations remain- 
ing unreduced, and so the planetary nature of the object 
was not recognized. 

In September of the same year, Le Yerrter wrote to 
Dr. Galle, then Assistant at the Observatory of Berlin, 
asking him to search for the new planet, and directing 
him to the place where it should be found. By the aid 
of an excellent star chart of this region, which had just 
been completed by Dr. Bremiker, the planet was found 
September 23d, 1846. 

The strict rights of discovery la}' with Le Yerrier, 
but the common consent of mankind has always credited 
Adams with an equal share in the honor attached to this 
most brilliant achievement. Indeed, it was only by the 
most unfortunate succession of accidents that the discovery 



368 



ASTRONOMY, 



did not attach to Adams' researches, 
fairness be said, and that is that the 



One thing must in 
results of Le Ver- 



ifier 



, which were reached after a most thorough investi- 



gation of the whole ground, were announced with an en- 
tire confidence, which, perhaps, was lacking in the other 
case. 

This brilliant discovery created more enthusiasm than 
even the discovery of Uranus, as it was by an exercise of 
far higher qualities that it was achieved. It appeared to 
.savor of the marvellous that a mathematician could say 



Fig. 98. 

to a working astronomer that by pointing his telescope to 
a certain small area, within it should be found a new 
major planet. Yet so it was. 

The general nature of the disturbing force which re- 
vealed the new planet may be seen by Fig. 98, which 
shows the orbits of the two planets, and their respective 
motions between 1781 and 1840. The inner orbit is that 
of Uranus, the outer one that of Neptune. The arrows 
passing from the former to the latter show the directions 
of the attractive force of Neptune. It will be seen that 



SATELLITE OF NEPTUNE. 369- 

the two planets were in conjunction in the year 1822. 
Since that time Uranus has, by its more rapid motion, 
passed more than 90° beyond JVeptune, and will continue 
to increase its distance from the latter until the begin- 
ning of the next century. 

Our knowledge regarding Neptune is mostly confined 
to a few numbers representing the elements of its motion. 
Its mean distance is more than 4,000,000,000 kilometres 
(2,775,000,000 miles) ; its periodic time is 164-78 years ; 
its apparent diameter is 2" -6 seconds, corresponding to a 
true diameter of 55,000 kilometres. Gravity at its surface 
is about nine tenths of the corresponding J e -restrial surface 
gravity. Of its rotation and physical condition nothing 
is known. Its color is a pale gieenish blue. It is attend- 
ed by one satellite, the elements of whose orbit are given 
herewith. It was discovered by IVlr. Lassell, of Eng- 
land, in 1847. It is about as faint as the two outer satel- 
lites of Uranus, and requires a telescope of twelve inches 
aperture or upward to be well seen. 

Elements of the Satellite op Neptune, from Washington 
Observations. 

Mean Daily Motion 61° -25679 

Periodic Time 5 d -87690 

Distance (log. A = 1-47814) 16" -275 

Inclination of Orbit to Ecliptic 145° 7' 

Longitude of Node (1850) 184° 30' 

Increase in 100 Years 1° 24' 



The great inclination of the orbit shows that it is turned nearly 
upside down ; the direction of motion is therefore retrogade. 



CHAPTER XL 

THE PHYSICAL CONSTITUTION OP THE 
PLANETS. 

It is remarkable that the eight large planets of the solar 
system, considered with respect to their physical constitu- 
tion as revealed by the telescope and the spectroscope, 
may be divided into four pairs, the planets of each pair 
having a great similarity, and being quite different from 
the adjoining pair. Among the most complete and sys- 
tematic studies of the spectra of all the planets are those 
made by Mr. Huggins, pf London, and Dr. Vogel, of 
Berlin. In what we have to say of the results of spectro- 
scopy, we shall depend entirely upon the reports of these 
observers. 

Mercury and Venus. — Passing outward from the sun. 
the first pair we encounter will be Mercury and Venus. 
The most remarkable feature of these two planets is a neg- 
ative rather than a positive one, being the entire absence 
of any certain evidence of change on their surfaces. We 
have already shown that Venus has a considerable atmos- 
phere, while there is no evidence of any such atmosphere 
around Mercury. They have therefore not been proved 
alike in this respect, yet, on the other hand, they have not 
been proved different. In every other respect than this, 
the similarity appears perfect. No permanent markings 
have ever been certainly seen on the disk of either. If, 
as is possible, the atmosphere of both planets is filled with 
clouds and vapor, no change, no openings, and no for- 



PHYSICAL CONSTITUTION' OF THE PLANETS. 371 

mations among these cloud masses are visible from the 
earth. Whenever either of these planets is in a certain 
position relative to the earth and the snn, it seemingly 
presents the same appearance, and not the slightest 
change occurs in that appearance from the rotation of the 
planet on its axis, which every analogy of the solar sys- 
tem leads us to believe must take place. 

When studied with the spectroscope, the spectra of 
Mercury and Venus do not differ strikingly from that of 
the sun. This would seem to indicate that the atmos- 
pheres of these planets do not exert any decided absorption 
upon the rays (. . light which pass through them ; or, at 
least, they absorb only the same rays which are absorbed 
by the atmosphere of the sun and by that of the earth. 
The one point of difference which Dr. Vogel brings out 
is, that the lines of the spectrum produced by the absorp- 
tion of our own atmosphere appear darker in the spectrum 
of Venus. If this were so, it would indicate that the at- 
mosphere of Venus is similar in constitution to that of 
our earth, .because it absorbs the same rays. But the 
means of measuring the darkness of the lines are as yet 
so imperfect that it is impossible to speak with certainty 
on a point like this. Dr. Yogel thinks that the light 
from Venus is for the most part reflected from clouds in 
the higher region of the planet's atmosphere, and there- 
fore reaches us without passing through a great depth of 
that atmosphere. 

The Earth and Mars. — These planets are distinguished 
from all the others in that their visible surfaces are marked 
by permanent features, which show them to be solid, and 
which can be seen from the other heavenly bodies. It is 
true that we cannot study the earth from any other body, 
but we can form a very correct idea how it would look if 
seen in this way (from the moon, for instance). Wherever 
the atmosphere was clear, the outlines of the continents 
and oceans would be visible, while they would be invisible 
where the air was cloudy. 



372 ASTRONOMT. 

Now, so far as we can judge from observations -made 
at so great a distance, never much less than forty mil- 
lions of miles, the planet Mars presents to our tele- 
scopes very much the same general appearance that the 
earth would if observed from an equally great distance. 
The only exception is that the visible surface of Mars is 
seemingly much less obscured by clouds than that of the 
earth would be. In other words, that planet has a more 
sunny sky than ours. It is, of course, impossible to say 
what conditions we might find could we take a much 
closer view of Mars : all we can assert is, that so far as 
we can judge from this distance, its surface is like that of 
the earth. 

This supposed similarity is strengthened by the spectro- 
scopic observations. The lines of the spectrum due to 
aqueous vapor in our atmosphere are found by Dr. Vogei* 
to be so much stronger in Mars as to indicate an absorp- 
tion by such vapor in its atmosphere. Dr. IIuggins had 
previously made a more decisive observation, having 
found a well-marked line to which there is no correspona- 
ing strong line in the solar spectrum. This would indi- 
cate that the atmosphere of Mars contains some element 
not found in our own, but the observations are too diffi- 
cult to allow of any well-established theory being yet 
built upon them. 

Jupiter and Saturn. — The next pair of planets are 
Jupiter and Saturn. Their peculiarity is that no solid 
crust or surface is visible from without. In this respect 
they differ from the earth and Mars, and resemble Mer- 
cury and Venus. But they differ from the latter in the 
very important point that constant changes can be seen 
going on at their surfaces. The nature of these changes 
has been discussed so fully in treating of these planets in- 
dividually, that we need not go into it more fully at pres- 
ent. It is sufficient to say that the preponderance of evi- 
dence is in favor of the view that these planets have no 
solid crusts whatever, but consist of masses of molten 



PHYSICAL CONSTITUTION OF THE PLANETS. 373 

matter, surrounded by envelopes of vapor constantly rising 
from the interior. 

The view that the greater part of the apparent volume of 
these planets is made of a seething mass of vapor is further 
strengthened by their very smaii specific gravity. This 
can be accounted for by supposing that the liquid interior 
is nothing more than a comparatively small central core, 
and that the greater part of the bulk of each planet is 
composed of vapor of small density. 

That the visible surfaces of Jupiter and Saturn are cov- 
ered by some kind of an atmosphere follows not only from 
the motion of the cloud forms seen there, but from the 
spectroscopic observations of Huggins in 1864. He 
found visible absorption-bands near the red end of the 
spectrum of each of these planets. Yogel found a com- 
plete similarity between the spectra of the two planets, 
the most marked feature being a dark band in the red. 
What is worthy of remark, though not at all surprising, is 
that this band is not found in the spectrum of Saturn's 
rings. This is what we should expect, as it is hardly pos- 
sible that these rings should have any atmosphere, owing 
to their very small mass. An atmosphere on bodies of so 
slight an attractive power would expand away by its own 
elasticity and be all attracted around the planet. 

Uranus and Neptune. — These planets have a strikingly 
similar aspect when seen through a telescope. They 
differ from Jupiter and Saturn in that no changes or va- 
riations of color or aspect can be made out upon their sur- 
faces ; and from the earth and Mars in the absence of any 
permanent features. Telescopically, therefore, we might 
classify them with Mercury and Venus, but the spectro- 
scope reveals a constitution entirely different from that of 
any other planets. The most marked features of their 
spectra are very dark bands, evidently produced by the 
absorption of dense atmospheres. Owing to the extreme 
faintness of the light which reaches us from these distant 
bodies, the regular lines of the solar spectrum are entirety 



374 



ASTRONOMY. 



invisible in their spectra, yet these dark bands which are 
peculiar to them have been seen by Huggins, Secchi, 

Vogel, and perhaps others. 

This classification of the 
eight planets into pairs is ren- 
dered yet more striking by 
the fact that it applies to 
what we have been able to 
discover respecting the rota- 
tions of these bodies. The 
rotation of the inner pair, 
Mercury and Venus, has 
eluded detection, notwith- 
standing their comparative 
proximity to us. The next 
pair, the earth and Mars, 
have perfectly definite times 
of rotation, because their 
outer surfaces consist of solid 
crusts, every part of which 
must rotate in the same time. 
The next pair, Jupiter and 
Saturn, have well-established 
times of rotation, but these 
times are not perfectly defi- 
nite, because the surfaces of 
these planets are not solid, 
and different portions of their 
mass may rotate in slightly 
different times. Jupiter and 
Fig. 99.— spectrum op uranus. Saturn have also in common 
a very rapid rate of rotation. Finally, the outer pair, Ura- 
nus and Neptune, seem to be surrounded by atmospheres of 
such density that no evidence of rotation can be gathered. 
Thus it seems that of the eight planets, only the central 
four have yet certainly indicated a rotation on their axes. 



CHAPTER XII. 

METEOES. 
% 1. PHENOMENA AND CAUSES OP METEORS. 

During the present century, evidence has been collected 
that countless masses of matter, far too small to be seen 
with the most powerful telescopes, are moving through 
the planetary spaces. This evidence is afforded by the 
phenomena of " aerolites, " " meteors," and "shooting 
stars." Although these several phenomena have been ob- 
served and noted from time to time since the earliest his- 
toric era, it is only recently that a complete explanation 
has been reached. 

Aerolites. — Reports of the falling of large masses of 
stone or iron to the earth have been familiar to antiqua- 
rian students for many centuries. Arago has collected 
several hundred of these reports. In one instance a monk 
was killed by the fall of one of these bodies. One or two 
other cases of death from this cause are supposed to have 
occurred. Notwithstanding the number of instances on 
record, aerolites fall at such wide intervals as to be ob- 
served by very few people, consequently doubt was fre- 
quently cast upon the correctness of the narratives. The 
problem where such a body could come from, or how it 
could get into the atmosphere to fall down again, formerly 
seemed so nearly incapable of solution that it required 
some credulity to admit the facts. When the evidence 
became so strong as to be indisputable, theories of their 
origin began to be propounded. One theory quite fashion- 



376 ASTRONOMY. 

able in the early part of this century was that they were 
thrown from volcanoes in the moon. This theory, 
though the subject of mathematical investigation by La 
Place and others, is now no longer thought of. 

The proof that aerolites did really fall to the ground 
first became conclusive by the fall being connected with 
other more familiar phenomena. Nearly every one who 
is at all observant of the heavens is familiar with bolides, 
or tire-balls — brilliant objects having the appearance of 
rockets, which are occasionally seen moving with great ve- 
locity through the upper regions of the atmosphere. 
Scarcely a year passes in which such a body of extraordi- 
nary brilliancy is not seen. Generally these bodies, bright 
though they may be, vanish without leaving any trace, or 
making themselves evident to any sense but that of sight. 
But on rare occasions their appearance is followed at an 
interval of several minutes by loud explosions like the dis- 
charge of a battery of artillery. On still rarer occasions, 
masses of matter fall to the ground. It is now fully 
understood that the fall of these aerolites is always ac- 
companied by light and sound, though the light may be 
invisible in the daytime. 

When chemical analysis was applied to aerolites, they 
were proved to be of extramundane origin, because they 
contained chemical combinations not found in terrestrial 
substances. It is true that they contained no new chemi- 
cal elements, but only combination of the elements which 
are found on the earth. These combinations are now so 
familiar to mineralogists that they can distinguish an 
aerolite from a mineral of terrestrial origin by a careful 
examination. One of the largest components of these 
bodies is iron. Specimens having very much the appear- 
ance of great masses of iron are found in the National 
Museum at Washington. 

Meteors. — Although the meteors we have described are 
of dazzling brilliancy, yet they run by insensible grada- 
tions into phenomena, which any one can see on any clear 



CAUSE OF METEORS. 377 

night. The most brilliant meteors of all are likely to be 
seen by one person only two or three times in his life. 
Meteors having the appearance and brightness of a distant 
rocket may be seen several times a year by any one in the 
habit of walking out during the evening and watching the 
sky. Smaller ones occur more frequently ; and if a care- 
ful watch be kept, it will be found that several of the 
faintest class of all, familiarly known as shooting stars, can 
be seen on every clear night. We can draw no distinction 
between the most brilliant meteor illuminating the whole 
sky, and perhaps making a noise like thunder, and the 
faintest shooting star, except one of degree. There seems 
to be every gradation between these extremes, so that all 
should be traced to some common cause. 

Cause of Meteors. — There is now no doubt that all these 
phenomena have a common origin, being due to the earth 
encountering innumerable small bodies in its annual course 
around the sun. The great difficulty in connecting mete- 
ors with these invisible bodies arises from the brilliancy 
and rapid disappearance of the meteors. The question 
may be asked why do they burn with so great an evolu- 
tion of light on reaching our atmosphere ? To answer this 
question, we must have recourse to the mechanical theory 
of heat. It is now known that heat is really a vibratory 
motion in the particles of solid bodies and a progressive 
motion in those of gases. By making this motion more 
rapid, we make the body warmer. By simply blowing air 
against any combustible body with sufficient velocity, it 
can be set on fire, and, if incombustible, the body will be 
made red-hot and finally melted. Experiments to deter- 
mine the degree of temperature thus produced have been 
made by Sir William Thomson, who finds that a veloci- 
ty of about 50 metres per second corresponds to a rise of 
temperature of one degree Centigrade. From this the 
temperature due to any velocity can be readily calculated 
on the principle that the increase of temperature is pro- 
portional to the " energy" of the particles, which again 



378 ASTRONOMY. 

is proportional to the square of trie velocity. Hence a 
velocity of 500 metres per second would correspond to a 
rise of 100° above the actual temperature of the air, so 
that if the latter was at the freezing-point the body would 
be raised to the temperature of boiling water. A velocity 
of 1500 metres per second would produce a red heat. This 
velocity is, however, much higher than any that we can 
produce artificially. 

The earth moves around the sun with a velocity of 
about 30,000 metres per second ; consequently if it met a 
body at rest the concussion between the latter and the at- 
mosphere would correspond to a temperature of more than 
300,000°. This would instantly dissolve any known sub- 
stance. 

As the theory of this dissipation of a body by moving 
with planetary velocity through the upper regions of our 
air is frequently misunderstood, it is necessary to explain 
two or three points in connection with it. 

(1.) It must be remembered that when we speak of 
these enormous temperatures, we are to consider them as 
potential, not actual, temperatures. We do not mean 
that the body is actually raised to a temperature of 300,- 
000°, but only that the air acts upon it as if it were put 
into a furnace heated to this temperature — that is, it is 
rapidly destroyed by the intensity of the heat. 

(2.) This potential temperature is independent of the 
density of the medium, being the same in the rarest as in 
the densest atmosphere. But the actual effect on the 
body is not so great in a rare as in a dense atmosphere. 
Every one knows that he can hold his hand for some time 
in air at the temperature of boiling water. The rarer the 
air the higher the temperature the hand would bear without 
injury. In an atmosphere as rare as ours at the height of 
50 miles, it is probable that the hand could be held for an 
indefinite period, though its temperature should be that 
of red-hot iron ; hence the meteor is not consumed so rap- 
idly as if it struck a dense atmosphere with planetary 



CAUSE OF METEORS. 379 

velocity. In the latter case it would probably disappear 
like a flash of lightning. 

(3.) The amount of heat evolved is measured not by that 
which would result from the combustion of the body, but 
by the vis viva (energy of motion) which the body loses in 
the atmosphere. The student of physics knows that mo- 
tion, when lost, is changed into a definite amount of 
heat. If we calculate the amount of heat which is equiv- 
alent to the energy of motion of a pebble having a veloc- 
ity of 20 miles a second, we shall And it sufficient to raise 
about 1300 times the pebble's weight of water from the 
freezing to the boiling point. This is many times as much 
heat as could result from burning even the most combusti- 
ble body. 

(4.) The detonation which sometimes accompanies the 
passage of very brilliant meteors is not caused by an ex- 
plosion of the meteor, but by the concussion produced by 
its rapid motion through the atmosphere. This concus- 
sion is of much the same nature as that produced by a 
flash of lightning. The air is suddenly condensed in front 
of the meteor, while a vacuum is left behind it. 

The invisible bodies which produce meteors in the way 
just described have been called meteo7*oids. Meteoric 
phenomena depend very largely upon the nature of the 
meteoroids, and the direction and velocity with which 
the} 7 are moving relatively to the earth. With very rare 
exceptions, they are so small and fusible as to be entirely 
dissipated in the upper regions of the atmosphere. Even 
of those eo hard and solid as to produce a brilliant light 
and the loudest detonation, only a small proportion reach 
the earth. It has sometimes happened that the meteoroid 
only grazes the atmosphere, passing horizontally through 
its higher strata for a great distance and continuing its 
course after leaving it. On rare occasions the body is so 
hard and massive as to reach the earth without being en- 
tirely consumed. The potential heat produced by its 
passage through the atmosphere is then all expended in 



380 ASTRONOMY. 

melting and destroying its outer layers, the inner nucleus 
remaining unchanged. When such a body first strikes 
the denser portion of the atmosphere, the resistance be- 
comes so great that the body is generally broken to pieces. 
Hence we very often find not simply a single aerolite, 
bat a small shower of them. 

Heights of Meteors. — Many observations have been 
made to determine the height at which meteors are seen. 
This is effected by two observers stationing themselves 
several miles apart and mapping out the courses of such 
meteors as they can observe. In order to be sure that the 
same meteor is seen from both stations, the time of each 
observation must be noted. In the case of very brilliant 
meteors, the path is often determined with considerable 
precision by the direction in which it is seen by accidental 
observers in various regions of the country over which it 
passes. 

The general result from numerous observations and in- 
vestigations of this kind is that the meteors and shooting 
stars commonly commence to be visible at a height of 
about 160 kilometres, or 100 statute miles. The separate 
results of course vary widely, but this is a rough mean of 
them. They are generally dissipated at about half this 
height, and therefore above the highest atmosphere which 
reflects the rays of the sun. From this it may be inferred 
that the earth's atmosphere rises to a height of at least 
1 60 kilometres. This is a much greater height than it was 
formerly supposed to have. 

§ 2. METEORIC SHOWERS. 

As already stated, the phenomena of shooting stars may 
be seen by a careful observer on almost any clear night. 
In general, not more than three or four of them will be 
seen in an hour, and these will be so minute as hardly to 
attract notice. But they sometimes fall in such numbers 
as to present the appearance of a meteoric shower. On 



METEORIC SHOWERS. 381 

rare occasions the shower lias been so striking as to till the 
beholders with terror. The ancient and mediaeval records 
contain many accounts of these phenomena which have 
been brought to light through the researches of antiqua- 
rians. The following is quoted by Professor Newton 
from an Arabic record : 

" In the year 599, on the last day of Moharrem, stars shot hither 
and thither, and flew against each other like a swarm of locusts ; 
this phenomenon lasted until daybreak ; people were thrown into 
consternation, and made supplication to the Most High : there was 
never the like seen except on the coming of the messenger of God, 
on whom be benediction and peace." 

It has long been known that some showers of this class 
occur at an interval of about a third of a century. One 
was observed by Humboldt, on the Andes, on the night 
of November 12th, 1799, lasting from two o'clock until 
daylight. A great shower was seen In this country in 
1833, and is well known to have struck the negroes of the 
Southern States with terror. The theory that the show- 
ers occur at intervals of 34 years was now propounded by 
Olbeks, who predicted a return of the shower in 1867. 
This prediction was completely fulfilled, but instead of ap- 
pearing in the year 1867 only, it was first noticed in 1866. 
On the night of November 13th of that year a remarkable 
shower was seen in Europe, while on the corresponding 
night of the year following it was again seen in this coun- 
try, and, in fact, was repeated for two or three years, grad- 
ually dying away. 

The occurrence of a shower of meteors evidently shows 
that the earth encounters a swarm of meteoroids. The 
recurrence at the same time of the year, w T hen the earth 
is in the same point of its orbit, shows that the earth 
meets the swarm at the same point in successive years. 
All the meteoroids of the swarm must of course be moving 
in the same direction, else they would soon be widely scat- 
tered. This motion is connected with the radiant poi?ht, 
a well-marked feature of a meteoric shower. 



382 ASTRONOMY. 

Badiant Point.— Suppose that, during a meteoric shower, we 
mark the patii of each meteor on a star map, as in the figure. If \* e 
continue the paths backward in a straight line, we shall find that 
they all meet near one and the same point of the celestial sphere— 
that is, they move as if they all radiated from this point. The 




RADIANT POINT OF METEORIC SHOWER. 



latter is, therefore, called the radiant point. In the figure the lines 
do not all pass accurately through the same point. This is owing 
to the unavoidable errors made in marking out the path. 

It is found that the radiant point is always in the same position 
among the stars, wherever the observer may be situated, and that 



M.ETEORH AJXD COMETS 383 

it does not partake of the diurnal motion of the earth — that is, as 
the stars apparently move toward the west, the radiant point moves 
with them. 

The radiant point is due to the fact that the meteoroids which 
strike the earth during a shower are all moving in the same direc- 
tion. If we suppose the earth to be at rest, and the actual motion 
of the meteoroids to be compounded with an imaginary motion 
equal and opposite to that of the earth, the motion of these imag- 
inary bodies will be the same as the actual relative motion of the 
meteoroids seen from the earth. These relative motions will all be 
parallel ; hence when the bodies strike our atmosphere the paths 
described by them in their passage will all be parallel straight 
lines. Now, by the principles of geometry of the sphere, a straight 
line seen by an observer at any point is projected as a great circle 
of the celestial sphere, of which the observer supposes himself to be 
the centre. If we draw a line from the observer parallel to the 
paths of the meteors, the direction of that line will indicate a point 
of the sphere through which all the paths will seem to pass ; this 
will, therefore, be the radiant point in a meteoric shower. 

A slightly different conception of the problem may be formed 
by conceiving the plane passing through the observer and contain- 
ing the path of the meteor. It is evident that the different planes 
formed by the parallel meteor paths will all intersect each other in 
a line drawn from the observer parallel to this path. This line 
will then intersect the celestial sphere in the radiant point. 

Orbits of Meteoric Showers. — From what has just been said, 
it will be seen that the position of the radiant point indicates the 
direction in which the meteoroids move relatively to the earth. If 
we also knew the velocity with which they are really moving in 
space, we could make allowance for the motion of the earth, and 
thus determine the direction of their actual motion in space. It 
will be remembered that, as just explained, the apparent or rela- 
tive motion is made up of two components — the one the actual 
motion of the body, the other the motion of the earth taken in an 
opposite direction. We know the second of these components 
already ; and if we know the velocity relative to the earth and the 
direction as given by the radiant point, we have given the resultant 
and one component in magnitude and direction. The computation 
of the other component is one of the simplest problems in kine- 
matics. Thus we shall have the actual direction and velocity of 
the meteoric swarm in space. Having this direction and velocity, 
the orbit of the swarm around the sun admits of being calculated. 

Relations of Meteors and Comets. — The velocity of the 
meteoroids does not admit of being determined from ob- 
servation. One element necessary for determining the 
orbits of these bodies is, therefore, wanting. In the case 
of the showers of 1799, 1S33, and 1866, commonly called 
the November showers, this element is given by the time 



384 ASTRONOMY. 

of revolution around the sun. Since the showers occur at 
intervals of about a third of a century, it is highly prob- 
able this is the periodic time of the swarm around the sun. 
The periodic time being known, the velocity at any dis- 
tance from the sun admits of calculation from the theory 
of gravitation. Thus we have all the data for determining 
the real orbits of the group of meteors around the sun. 
The calculations necessary for this purpose were made 
by Le Yerrier and other astronomers shortly after the 
great shower of 1866. The following was the orbit as 
given by Le Yerrier : 

Period of revolution 33-25 years. 

Eccentricity of orbit • 9044. 

Least distance from the sun • 9890. 

Inclination 'of orbit 165° 19'. 

Longitude of the node 51° 18'. 

Position of the perihelion (near the node). 

The publication of this orbit brought to the attention 
of the world an extraordinary coincidence which had 
never before been suspected. In December, 1865, a 
faint telescopic comet was discovered by Tempel at Mar- 
seilles, and afterward by H. P. Tuttle at the Naval 
Observatory, Washington. Its orbit was calculated by 
Dr. Oppolzer, of Yienna, and his results were finally pub- 
lished on January 28th, 1867, in the Astronomische Naok- 
richten • they were as follows : 

Period of revolution 33-18 years. 

Eccentricity of orbit • 9054. 

Least distance from the sun • 9765. 

Inclination of orbit 162° 42'. 

Longitude of the node 51° 26'. 

Longitude of the perihelion 42° 24'. 

The publication of the cometary orbit and that of the 
orbit of the meteoric group were made independently with- 
in a few days of each other by two astronomers, neither 
of whom had any knowledge of the work of the other. 
Comparing them, the result is evident. The swarms of 
meteoroids which cause the November showers move in 
the same orbit with Tempel 's comet. 



THE AUGUST METEORS. 385 

Tempel's comet passed its perihelion in January, 
1866. The most striking meteoric shower commenced 
in the following November, and was repeated during 
several years. It seems, therefore, that the meteoroids 
which produce these showers follow after Tempel's comet, 
moving in the same orbit with it. This shows a curious 
relation between comets and meteors, of which we shall 
speak more fully in the next chapter. When this fact 
was brought out, the question naturally arose whether the 
same thing might not be true of other meteoric showers. 

Other Showers of Meteors — Although the November 
showers are the only ones so brilliant as to strike the ordi- 
nary eye, it lias long been known that there are other 
nights of the year in which more shooting stars than usual 
are seen, and in which the large majority radiate from one 
point of the heavens. This shows conclusively that they 
arise from swarms of meteoroids moving together around 
the sun. 

August Meteors. — The best marked of these minor 
showers .occurs about August 9th or 10th of each year. 
The radiant point is in the constellation Perseus. By 
watching the eastern heavens toward midnight on the 9th 
or 10th of August of any year, it will be seen that numer- 
ous meteors move from north-east toward south-west, hav- 
ing often the distinctive characteristic of leaving a trail 
behind, which, however, vanishes in a few moments. As- 
suming their orbits to be parabolic, the elements were cal- 
culated by Schiaparelli, of Milan, and, on comparing with 
the orbits of observed comets, it was found that these 
meteoroids moved in nearly the same orbit as the second 
comet of 1862. The exact period of this comet is not 
known, although the orbit is certainly elliptic. Accord- 
ing to the best calculation, it is 124: years, but for reasons 
given in the next chapter, it may be uncertain by ten 
years or more. 

There is one remarkable difference between the August and the 
November meteors. The latter, as we have seen, appear for two 



386 ASTRONOMY. 

or three consecutive years, and then are not seen again until about 
thirty years have elapsed. But the August meteors are seen every 
year. This shows that the stream of August meteoroids is endless, 
every part of the orbit being occupied by them, while in the case 
of the November ones they are gathered into a group. 

We may conclude from this that the November meteoroids have 
not been permanent members of our system. It is beyond all prob- 
ability that a group comprising countless millions of such bodies 
should all hav^e the same time of revolution. Even if they had the 
same time in the beginning, the different actions of the planets on 
different parts of the group would make the times different. The 
result would be that, in the course of ages, those which had the 
most rapid motion would go further and further ahead of the 
others until they got half a revolution ahead of them, and would 
tiually overtake those having the slowest motion. The swiftest and 
slowest one would then be in the position of two race-horses running 
around a circular track for so long a time that the swiftest horse 
has made a complete run more than the slowest one and has over- 
taken him from behind. When this happens, the meteoroids will 
be scattered all around the orbit, and we shall have a shower in 
November of every year. The fact that has not yet happened shows 
that they have been revolving for only a limited length of time, 
probably only a very few thousand years. 

Although the total mass of these bodies is very small, yet their 
number is beyond all estimation. Professor Newton has estimated 
that, taking the whole earth, about seven million shooting stars are 
encountered every twenty-four hours. This would make between 
two and three thousand million meteoroids which are thus, as it 
were, destroyed every year. But the number which the earth can 
encounter in a year is only an insignificant fraction of the total 
number, even in the solar system. It may be interesting to calculate 
the ratio of the space swept over by the earth in the course of a year 
to the volume of the sphere surrounding the sun and extending out 
to the orbit of Neptune. We shall find this ratio to be only as one 
to about three millions of millions. If we measure by the number 
of meteoroids in a cubic mile, we might consider them very thinly 
scattered. There is, in fact, only a single meteor to several million 
cubic kilometres of space in the heavens. Yet the total number 
is immensely great, because a globe including the orbit of Neptune 
would contain millions of millions of millions of millions of cubic 
kilometres.* If we reflect, in addition, that the meteoroids probably 

* The computations leading to this result may be made in the fol- 
lowing manner : 

I. To find the cubical space swept through by the earth in the course of 
a year. If we put tt for the ratio of the circumference of a circle to its 
diameter, and p for the radius of the earth, the surface of a plane section 
of the earth passing through its centre will be t p 2 . Multiplying this 
by the circumference of the earth's orbit, we shall have the space re- 
quired, which we readily find to be more than 30,000 millions of 
millions of kilometres. Since, in sweeping through this space, the 
earth encounters about 2500 millions of meteoroids, it follows that 



THE ZODIACAL LIGHT. 387 

weigh but a few grains each, we shall see how it is that they are en- 
tirely invisible even with powerful telescopes. 

The Zodiacal Light. — If we observe the western sky 
during the winter or spring months, about the end of the 
evening twilight, we shall see a stream of faint light, a 
little like the Milky Way, rising obliquely from the west, 
and directed along the ecliptic toward a point south-west 
from the zenith. This is called the zodiacal light. It 
may also be seen in the east before daylight in the morn- 
ing during the autumn months, and has sometimes been 
traced all the way across the heavens. Its origin is still 
involved in obscurity, but it seems probable that it arises 
from an extremely thin cloud either of meteoroids or of 
semi-gaseous matter like that composing the tail of a 
comet, spread all around the sun inside the earth's orbit. 
The researches of Professor A. W. Wright show that its 
spectrum is probably that of reflected sunlight, a result 
which gives color to the theory that it arises from a cloud 
of meteoroids revolving round the sun. 

there is only one meteoroid to more than ten millions of cubic kil- 
ometres. 

II. To find the ratio of the sphere of space within the orbit of Neptune to 
the space swept through by the earth in a year. Let us put r for the dis- 
tance of the earth from the sun. Then the distance of Neptune may 
be taken as 30 r, and this will be the radius of the sphere. The cir- 
cumference of the eartu's orbit will than be 2 tt r, and the space swept 
over will be 2 tt 2 r p 2 . The sphere of Neptune will be 

1 7T 30 3 r* = 36,000 it r\ nearly. 

The ratio of the two spaces will be 

18,000 r 2 __ n r 2 

r— = 6,000 — —, nearly. 

w p 3 p 2 

r 
The ratio — is more than 23,000, showing the required ratio to be 

P 

about three millions of millions. The total number of scattered mete- 
oroids is therefore to be reckoned by millions of millions of millions. 



CHAPTER XIII. 

COMETS. 
§ 1. ASPECT OP COMETS. 

Comets are distinguished from the planets both by their 
aspects and their motions. They come into view without 
anything to herald their approach, continue in sight for a 
few weeks or months, and then gradually vanish in the 
distance. They are commonly considered as composed of 
three parts, the nucleus, the coma (or hair), and the tail. 

The nucleus of a comet is, to the naked eye, a point of 
light resembling a star or planet. Yiewed in a telescope, 
it generally has a small disk, but shades off so gradually 
that it is difficult to estimate its magnitude. In large 
comets, it is sometimes several hundred miles in diameter, 
but never approaches the size of one of the larger planets. 

The nucleus is always surrounded by a mass of foggy 
light, which is called the coma. To the naked eye, the 
nucleus and coma together look like a star seen through a 
mass of thin fog, which surrounds it with a sort of halo. 
The coma is brightest near the nucleus, so that it is hardly 
possible to tell where the nucleus ends and where the 
coma begins. It shades off in every direction so gradually 
that no definite boundaries can be fixed to it. The 
nucleus and coma together are generally called the head 
of the comet. 

The tail of the comet is simply a continuation of the 
coma extending out to a great distance, and always di- 
rected away from the sun. It has the appearance of a 
stream of milky light, which grows fainter and broader 



A8PEGT OF COMETS. 389 

as it recedes from the head. Like the coma, it shades off 
so gradually that it is impossible to fix any boundaries to 
it. The length of the tail varies from 2° or 3° to 90° or 
more. Generally the more brilliant the head of the comet, 
the longer and brighter is the tail. It is also often brighter 
and more sharply defined at one edge than at the other. 

The above description applies to comets which can be 
plainly seen by the naked eye. After astronomers began 
to sweep the heavens carefully with telescopes, it was 
found that many comets came into sight which would 
entirely escape the unaided vision. These are called tel- 
escopic comets. Sometimes six or more of such comets are 
discovered in a single year, while one of the brighter class 
may not be seen for ten years or more. 





Fig. 101. — telescopic comet with- Fig. 102.— telescopic comet 
out a nucleus. with a nucleus. 

When comets are studied with a telescope, it is found 
that they are subject to extraordinary changes of structure. 
To understand these changes, we must begin by saying that 
comets do not, like the planets, revolve around the sun in 
nearly circular orbits, but always in orbits so elongated 
that the comet is visible in only a very small part of its 
course. When one of these objects is first seen, it is gen- 
erally approaching the sun from the celestial spaces. 
At this time it is nearly always devoid of a tail, and some- 
times of a nucleus, presenting the aspect of a thin patch 
of cloudy light, which may or may not have a nucleus in 



390 ASmONOMY. 

its centre. As it approaches the sun, it is generally seen 
to grow brighter at some one point, and there a nucleus 
gradually forms, being, at first, so faint that it can scarcely 
be distinguished from the surrounding nebulosity. The 
latter is generally more extended in the direction of the 
sun, thus sometimes giving rise to the erroneous impres- 
sion of a tail turned toward the sun. Continuing the 
watch, the true tail, if formed at all, is found to begin 
very gradually. At first so small and faint as to be almost 
invisible, it grows longer and brighter every day, as long 
as the comet continues to approach the sun. 



§ 2. THE VAPOROUS ENVELOPES. 

If a comet is very small, it may undergo no changes of 
aspect, except those just described. If it is an unusually 
bright one, the next object noticed by telescopic examina- 
tion will be a bow surrounding the nucleus on the side 
toward the sun. This bow will gradually rise up and 
spread out on all sides, finally assuming the form of a 
semicircle having the nucleus in its centre, or, to speak 
with more precision, the form of a parabola having the 
nucleus near its focus. The two ends of this parabola 
will extend out further and further so as to form a part 
of the tail, and finally be lost in it. Continuing the 
watch, other bows will be found to form around the nu- 
cleus, all slowly rising from it like clouds of vapor. 
These distinct vaporous masses are called the envelopes : 
they shade off gradually into the coma so as to be with 
difficulty distinguished from it, and indeed may be con- 
sidered as part of it. The inner envelope is sometimes 
connected with the nucleus by one or more fan -shaped 
appendages, the centre of the fan being in the nucleus, 
and the envelope forming its round edge. This appear- 
ance is apparently caused by masses of vapor streaming 
up from that side of the nucleus nearest the sun, and grad- 
ually spreading around the comet on each side. The 



ENVELOPES OF COMETS. 391 

form of a bow is not the real form of the envelopes, but 
only the apparent one in which we see them projected 
against the background of the sky. Their true form is 
similar to that of a paraboloid of revolution, surrounding 
the nucleus on all sides, except that turned from the sun. 
It is, therefore, a surface and not a line. Perhaps its form 
can be best imagined by supposing the sun to be directly 
above the comet, and a fountain, throwing a liquid hori- 
zontally on all sides, to be built upon that part of the 
comet which is uppermost. Such a fountain would throw 
its water in the form of a sheet, falling on all sides of the 
cometic nucleus, bat not touching it. Two or three vapor 
surfaces of this kind are sometimes seen around the comet, 
the outer one enclosing each of the inner ones, but no two 
touching each other. 




Fig. 103. — formation op envelopes. 

To give a clear conception of the formation and motion of the 
envelopes, we present two figures. The first of these shows the ap- 
pearance of the envelopes in four successive stages of their course, 
and may be regarded as sections of the real umbrella-shaped sur- 
faces which they form. In all these figures, the sun is supposed to 
be above the comet in the figure, and the tail of the comet to be 
directed downward. In a the sheet of vapor has just begun to 
rise. In Z> it is risen and expanded yet further. In c it has begun 
to move away and pass around the comet on all sides. Finally, 
in d this last motion has gone so far that the higher portions 
have nearly disappeared, the larger part of the matter having 
moved away toward the tail. Before the stage c is reached, a 
second envelope will commonly begin to rise as at a, so that two 
or three may be visible at the same time, enclosed within each 
other. 

In the next figure the actual motion of the matter compos- 



392 ASTRONOMY. 

ing the envelopes is shown by the courses of the several dotted 
lines. This motion, it will be seen, is not very unlike that of 
water thrown up from a fountain on the part of the nucleus 
nearest the sun and then falling down on all sides. The point in 
which the motion of the cometic matter differs from that of the 
fountain is that, instead of being thrown in continuous streams, 
the action is intermittent, the fountain throwing up successive 
sheets of matter instead of continuous streams. 

From the gradual expansion of these envelopes around the head 
of the comet an I the continual formation of new ones in the im- 
mediate neighborhood of the nucleus, they would seem to be due 
to a process of evaporation going on from the surface of the latter. 
Each layer of vapor thus formed rises up and spreads out con- 
tinually until the part next the sun attains a certain maximum 
height. Then it gradually moves away from the sun, keeping its 
distance from the comet, at least until it passes the latter on every 
side, and continues onward to form the tail. 




Fig. 104. — formation of comet's tail. 

These phenomena were fully observed in the great comet of 
1858, the observations of which were carefully collected by the 
late Professor Bond, of Cambridge. The envelopes of this comet 
were first noticed on September 20th, when the outer one was 16* 
above the nucleus and the inner one 3". The outer one disap- 
peared on September 30th at a height of about 1'. In the mean 
while, however, a third had appeared, the second having gradually 
expanded so as to take the place of the first. Seven successive 
envelopes in all were seen to rise from this comet, the last one com- 
mencing on October 20th, when all the others had been dissipated. 
The rate at which the envelopes ascended was generally from 50 to 
60 kilometres per hour, the ordinary speed of a railway-train. 

The first one rose to a height of about 30,000 kilometres, but it 
was finally dissipated. But the successive ones disappeared at a 
lower and lower elevation, the sixth being lost sight of at a height 
of about 10,000 kilometres. 



SPECTRA OF COMETS. 



393 



In the great comet of 1861, eleven envelopes were seen between 
July 2d, when portions of three were in sight, and the 19th of 
the same month, a new one rising at regular intervals of every sec- 
ond day. Their evolution and dissipation were accomplished with 
much greater rapidity than in the case of the great comet of 1858, 
an envelope requiring but two or three days instead of two or three 
weeks to pass through all its phases. 



§ 3. THE PHYSICAL CONSTITUTION OP COMETS. 

To tell exactly what a comet is, we should be able to 
show how all the phenomena it presents would follow from 
the properties of matter, as we learn them at the surface 
of the earth. This, however, no one has been able to do, 
many of the phenomena being such as we should not ex- 
pect from the known constitution of matter. All we can 
do, therefore, is to present the principal characteristics of 
comets, as shown by observation, and to explain what is 
wanting to reconcile these characteristics with the known 
properties of matter. 

In the first place, all comets which have been examined 
with the spectroscope show a spectrum composed, in part 
at least, of bright lines or bands. These lines have been 
supposed to be identified with those of carbon ; but 
although the similarity of aspect is very striking, the iden- 
tity cannot be regarded as proven. 




Fig. 105.— spectra of olefiant gas and of a comet. 

In the annexed figure the upper speetrum, A, is that of carbon 
taken in olefiant gas, and the lower one, B, that of a comet. These 
spectra interpreted in the usual way would indicate, firstly, that 
the comet is gaseous ; secondly, that the gases which compose it 
are so hot as to shine by their own light. But we cannot admit 



394 ASTRONOMY. 

these interpretations without bringing in some additional theory. 
A mass of gas surrounding so minute a body as the nucleus of a 
telescopic comet would expand into space by virtue of its own 
elasticity unless it were exceedingly raie. Moreover, if it were 
incandescent, it would speedily cool off so as to be no longer self- 
luminous. We must, therefore, propose some theory to account 
for the continuation of the luminosity through many centuries, 
such as electric activity or phosphorescence. But without further 
proof of action of these causes we cannot accept their reality. We 
are, therefore, unable to say with certainty how the light in the 
spectrum of comets which produces the bright lines has its origin. 

In the last chapter it was shown that swarms of minute 
particles called meteoroids follow certain comets in their 
orbits. This is no doubt true of all comets. We can only 
regard these meteoroids as fragments or debris of the 
comet. The latter has therefore been considered by Pro- 
fessor Newton as made up entirely of meteoroids or small 
detached masses of matter. These masses are so small and 
so numerous that they look like a cloud, and the light 
which they reflect to our eyes has the milky appearance 
peculiar to a comet. On this theory a telescopic comet 
which has no nucleus is simply a cloud of these minute 
bodies. The nucleus of the brighter comets may either 
be a more condensed mass of such bodies or it may be a 
solid or liquid body itself. 

If the reader has any difficulty in reconciling this theory 
of detached particles with the view already presented, 
that the envelopes from which the tail of the comet is 
formed consist of layers of vapor, he must remember that 
vaporous masses, such as clouds, fog, and smoke, are 
really composed of minute separate particles of water or 
carbon. 

Formation of the Comet's Tail. — The tail of the comet 
is not a permanent appendage, but is composed of the 
masses of vapor which we have already described as as- 
cending from the nucleus, and afterward moving away 
from the sun. The tail which we see on one evening is 
not absolutely the same we saw the evening before, a 



MOTIONS OF COMETS. 395 

portion of the latter having been dissipated, while new 
matter has taken its place, as with the stream of smoke from 
a steamship. The motion of the vaporous matter which 
forms the tail being always away from the sun, there 
seems to be a repulsive force exerted by the sun upon it. 
The form of the comet's tail, on the supposition that it is 
composed of matter thus driven away from the sun with 
a uniformly accelerated velocity, has been several times 
investigated, and found to represent the observed form of 
the tail so nearly as to leave little doubt of its correctness. 
We may, therefore, regard it as an observed fact that the 
vapor which rises from the nucleus of the comet is repelled 
by the sun instead of being attracted toward it, as other 
masses of matter are. 



No adequate explanation of this repulsive force has ever been 
given. It has, indeed, been suggested that it is electrical in its 
character, but no one has yet proven experimentally that the attrac- 
tion exerted by the sun upon terrestrial bodies is influenced by their 
electrical state. If this were done, we should have a key to one of 
the most difficult problems connected with the constitution of 
comets. As the case now stands, the repulsion of the sun upon the 
comet's tail is to be regarded as a well-ascertained and entirely 
isolated fact which has no known counterpart in any other observed 
fact of nature. 

In view of the difficulties we find in explaining the phenomena of 
comets by principles based upon our terrestrial chemistry and 
physics, the question will arise whether the matter which composes 
these bodies may not be of a constitution entirely different from 
that of any matter we are acquainted with at the earth's surface. 
If this were so, it would be impossible to give a complete explanation 
of comets until we know what forms matter might possibly assume 
different from those we find it to have assumed in our labora- 
tories. This is a question which we merely suggest without 
attempting to speculate upon it. It can be answered only by ex- 
perimental researches in chemistry and physics. 



§ 4. MOTIONS OF COMETS. 

Previous to the time of Newton, no certain knowledge 
respecting the actual motions of comets in the heavens 
had been acquired, except that they did not move around 



396 



ASTRONOMY 



the sun like the planets. When Newton investigated the 
mathematical results of the theory of gravitation, he found 
that a body moving under the attraction of the sun might 
describe either of the three conic sections, the ellipse, par- 
abola, or hyperbola. Bodies moving in an ellipse, as the 
planets, would complete their orbits at regular intervals 
of time, according to laws already laid down. But if the 
body moved in a parabola or a hyperbola, it would never 
return to the sun after once passing it, but would move off 




Fig. 106.— elliptic and parabolic orbits. 

to infinity. It was, therefore, very natural to conclude 
that comets might be bodies which resemble the planets in 
moving under the sun's attraction, but which, instead of 
describing an ellipse in regular periods, like the planets, 
move in parabolic or hyperbolic orbits, and therefore 
only approach the sun a single time during their whole 
existence. 

This theory is now known to be essentially true for 



ORBITS OF COMETS. 397 

most of the observed comets. A few are indeed found to 
be revolving around the sun in elliptic orbits, which differ 
from those of the planets only in being much more eccen- 
tric. But the greater number which have been observed 
have receded from the sun in orbits which we are unable 
to distinguish from parabolas, though it is possible they 
may be extremely elongated ellipses. Comets are there- 
fore divided with respect to their motions into two classes : 
(1) periodic comets, which are known to move in elliptic 
orbits, and to return to the sun at fixed intervals ; and (2) 
parabolic comets, apparently moving in parabolas, never 
to return. 

The first discovery of the periodicity of a comet was 
made by Halley in connection with the great comet of 
1682. Examining the records of observations, he found 
that a comet moving in nearly the same orbit with that of 
1682 had been seen in 1607, and still another in 1531. 
He was therefore led to the conclusion that these three 
comets were really one and the same object, returning to 
the sun at intervals of about 75 or 76 years. He there- 
fore predicted that it would appear again about the year 
1758. But such a prediction might be a year or more in 
error, owing to the effect of the attraction of the planets 
upon the comet. In the mean time the methods of calcu- 
lating the attraction of the planets were so far improved 
that it became possible to make a more accurate predic- 
tion. As the year 1759 approached, the necessary com- 
putations were made by the great French geometer Clai- 
kaut, who assigned April 13th, 1759, as the day on which 
the comet would pass its perihelion. This prediction 
was remarkably correct. The comet was first seen on 
Christmas-day, 175S, and passed its perihelion March 
12th, 1759, only one month before the predicted time. 
The comet returned again in 1835, within three days of 
the moment predicted by De Pontecoulant, the most 
successful calculator. The next return will probably take 



398 



ASTRONOMY. 



place in 1911 or 1912, the exact time being still unknown, 
because the necessary computations have not yet been 
made. 

We give a hgure showing the position of the orbit of 
Halley's comet relative to the orbits of the four outer 

planets. It attain- 
ed its greatest dis- 
tance from the sun, 
far beyond the or- 
bit of Neptune, 
about the year 18 73, 
and th e n com- 
menced its return 
journey. The fig- 
ure shows the prob- 
able position of the 
comet in 1874. It 
was then far be- 
yond the reach of 
the most powerful 
telescope, but its distance and direction admit of being 
calculated with so much precision that a telescope could 
be pointed at it at any required moment. 

We have already stated that great numbers of comets, 
too faint to be seen by the naked eye, are discovered by 
telescopes. A considerable number of these telescopic 
comets have been found to be periodic. In most cases, 
the period is many centuries in length, so that the comets 
have only been noticed at a single visit. Eight or 
nine, however, have been found to be of a period so short 
that they have been observed at two or more returns. 

We present a table of such of the periodic comets as 
have been actually observed at two or more returns. A 
number of others are known to be periodic, but have been 
observed only on a single visit to our system. 




Fig. 107. — okbit of halley's comet. 



ORBITS OF COMETS. 



399 





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400 ASTRONOMY. 

Theory of Cometary Orbits. — There is a property of all or- 
bits of bodies around the sun, an understanding of which will 
enable us to form a clear idea of some causes which affect the 
motion of comets. It may be expressed in the following theorem : 

The mean distance of a body from the sun, or the major semi-axis of 
the ellipse in which it revolves, depends only upon the velocity of the 
body at a given distance from the sun, and may be found by the 
formula, 



2 u — r v- ' 

in which r is the distance from the sun, v the velocity with which 
the body is moving, and ju. a constant proportional to the mass of 
the sun and depending on the units of time and length we adopt. 

To understand this formula, let us imagine ourselves in the celes- 
tial spaces, with no planets in our neighborhood. Suppose we have 
a great number of balls and shoot them out with the same velocity, 
but in different directions, so that they will describe orbits around the 
sun. Then the bodies will all describe different orbits, owing to 
the different directions in which we threw them, but these orbits 
will all possess the remarkable property of having equal major 
axes, and therefore equal mean distances from the sun. Since, by 
Kepler's third law, the periodic time depends only upon the 
mean distance, it follows that the bodies will have the same time 
of revolution around the sun. Consequently, if we wait patiently 
at the point of projection, they will all make a revolution in the 
same time, and will all come back again at the same moment, each 
one coming from a direction the opposite of that in which it was 
thrown. 

In the above formula the major axis is given by a fraction, having 

the expression 2 [i — rv" 2, for its denominator ; it follows that if the 

2 u 
square of the velocity is almost equal to — [-, the value of a will 

become very great, because the denominator of the fraction will be 
very small. If the velocity is such that 2 /u — r v 2 is zero, the mean 
distance will become infinite. Hence, in this case the body will 
fly off to an infinite distance from the sun and never return. 
Much less will it return if the velocity is still greater. Such a 
velocity will make the value of a algebraically negative and will 
correspond to the hyperbola. 

If we take one kilometre per second as the unit of velocity, and 
the mean distance of the earth from the sun as the unit of distance, 
the value of // will be represented by the number 875, so that the 

formula for a will be a = -— -. From this equation, we may 

1750— rv 2 n ' J 

calculate what velocity a body moving around the sun must have 

at any given distance r, in order that it may move in a parabolic 

orbit — that is, that the denominator of the fraction shall vanish. 

This condition will give v 2 = . At the distance of the earth 



ORIGIN OF COMETS. 401 

from the sun we have r = 1; so that, at that distance, v will be the 
square root of 1750, or nearly 42 kilometres per second. The fur- 
ther we get out from the sun, the less it will be ; and we may remark, 
as an interesting theorem, that whenever the comet is at the dis- 
tance of one of the planetary orbits, its velocity must be equal to 
that of the planet multiplied by the square root of 2, or 1*414, etc. 
Hence, if the velocity of any planet were suddenly increased by a 
little more than T 4 F of its amount, its orbit would be changed into 
a parabola, and it would fly away from the sun, never to return. 

It follows from all this that if the astronomer, by observing the 
course of a comet along its orbit, can determine its exact velocity 
from point to point, he can thence calculate its mean distance from 
the sun and its periodic time. But it is found that the velocity of 
a large majority of comets is so nearly equal to that required for 
motion in a parabola, that the difference eludes observation. It is 
hence concluded that most comets move nearly in parabolas, and 
will either never return at all or, at best, not until after the lapse of 
many centuries. 



§ 5. ORIGIN OP COMETS. 

All that we know of comets seems to indicate that they 
did not originally belong to our system, but became mem- 
bers of it through the disturbing forces of the planets. 
From what was said in the last section, it will be seen that 
if a comet is moving in a parabolic orbit, and its velocity 
is diminished at any point by ever so small an amount, its 
orbit will be changed into an ellipse ; for in order that the 
orbit may be parabolic, the quantity 2 \i—r v 1 must remain 
exactly zero. But if we then diminish v by the smallest 
amount, this expression will become finite ' and positive, 
and a will no longer be infinite. Now, the attraction of 
a planet may have either of two opposite effects ; it may 
either increase or diminish the velocity of the comet. 
Hence if the latter be moving in a parabolic orbit, the at- 
traction of a planet might either throw it out into a hyper- 
bolic orbit, so that it would never again return to the sun, 
but wander forever through the celestial spaces, or it 
might change its orbit into a more or less elongated ellipse. 

Suppose CD to represent a small portion of the orbit 
of the planet and A B a small portion of the orbit of a 
comet passing near it. Suppose also that the comet passes 



402 



ASTRONOMY. 



a little in front of the planet, and that the simultaneous 
positions of the two bodies are represented by the corre- 
sponding letters of the alphabet a, b, c, d, etc.; the shortest 
distance of the two bodies will be the line c c, and it is 
then that the attraction will be the most powerful. 
Between c c and d d the planet will attract the comet almost 
directly backward. It follows then that if a comet pass 
the planet in the way here represented, its velocity will be 
retarded by the attraction of the latter. If therefore it be 
a parabolic comet, the orbit will be changed into an 
ellipse. The nearer it passes to the planet, the greater 
will be the change, so long as it passes in front of it. If 

it passes behind, the 
reverse effect will 
follow, and the mo- 
tion will be accele- 
rated. The orbit will 
then be changed into 
a hyperbola. The or- 
bit finally described 
after the comet leaves 
our system will de- 
pend upon whether 
its velocity is accele- 
rated or retarded by 
the combined attraction of all the planets. 

All the studies which have been made of comets seem 
to show that they originally moved in parabolic orbits, and 
were brought into elliptic orbits in this way by the attrac- 
tion of some planet. The planet which has thus brought 
in the greatest number is no doubt Jupiter. In fact, the 
orbits of several of the periodic comets pass very near to 
that planet. It might seem that these orbits ought almost 
to intersect that of the planet which changed them. This 
would be true at first, but owing to the constant change in 
the position of the cometary orbit, produced by the at- 
traction of the planets, the orbits would gradually move 




Fig. 108.— attkaction op planet on 

COMET. 



ORIGIN OF COMETS. 403 

away from each other, so that in time there might be no 
approach whatever of the planet to the comet. 

A remarkable case of this sort was afforded by a comet 
discovered in June, 1770. It was observed in all nearly 
four months, and was for some time visible to the naked 
eye. On calculating its orbit from all the observations, 
the astronomers were astonished to find it to be an ellipse 
with a period of only "B.\e or six years. It ought therefore 
to have appeared again in 1776 or 1777, and should have 
returned to its perihelion twenty times before now, and 
should also have been visible at returns previouslo that at 
which it was first seen. But not only was it never seen 
before, but it has never been seen since ! The reason of 
its disappearance from view was brought to light on cal- 
culating its motions after its first discovery. At its re- 
turn in 1776, the earth was not in the right part of its 
orbit for seeing it. On passing out to its aphelion again, 
about the beginning of 1779, it encountered the planet 
Jupiter, and approached so near it that it was impossible 
to determine on which side it passed. This approach, it 
will be remembered, could not be observed, because the 
comet was entirely out of sight, but it was calculated with 
absolute certainty from the theory of the comet's motion. 
The attraction of Jupiter, therefore, threw it into some 
orbit so entirely different that it has never been seen since. 

It is also highly probable that the comet had just been 
brought in by the attraction of Jupiter on the very revo- 
lution in which it was first observed. Its history is this : 
Approaching the sun from the stellar spaces, probably for 
the first time, it passed so near Jupiter in 1767 that its or- 
bit was changed to an ellipse of short period. It made 
two complete revolutions around the sun, and in 1779 
again met the planet near the same place it had met him 
before. The orbit was again altered so much that no tel- 
escope has found the comet since. No other case so re- 
markable as this has ever been noticed. 

Not only are new comets occasionally brought in from 



404 ASTRONOMY. 

the stellar spaces, but old ones may, as it were, fade away 
and die. A case of this sort is afforded by Biela's comet, 
which has not been seen since 1852, and seems to have en- 
tirely disappeared from the heavens. Its history is so in 
structive that we present a brief synopsis of it. It was first 
observed in 1772, again in 1805, and then a third time in 
1826. It was not until this third apparition that its peri- 
odicity was recognized and its previous appearances iden- 
tified as those of the same body. The period of revolu- 
tion was found to be between six and seven years. It was 
so small as to be visible in ordinary telescopes only when 
the earth was near it, which would occur only at one re- 
turn out of three or four. So it was not seen again until 
near the end of 1845. Nothing remarkable was noticed in 
its appearance until January, 1846, when all were aston- 
ished to find it separated into two complete comets, one a 
little brighter than the other. The computation of Pro- 
fessor Hubbard makes the distance of the two bodies to 
have been 200,000 miles. 

The next observed return was that of 1852, when the 
two comets were again viewed, but far more widely 
separated, their distance having increased to about a mil- 
lion and a half of miles. Their brightness was so nearly 
equal that it was not possible to decide which should be 
considered the principal comet, nor to determine with 
certainty which one should be considered as identical with 
the comet seen during the previous apparition. 

Though carefully looked for at every subsequent return, 
neither comet has been seen since. In 1872, Mr. Pogson, 
of Madras, thought that he got a momentary view of the 
comet through an opening between the clouds on a stormy 
evening, but the position in which he supposed himself to 
observe it was so far from the calculated one that his obser- 
vation has not been accepted. 

Instead of the comet, however, we had a meteoric shower. 
The orbit of this comet almost intersects that of the earth. 
It was therefore to be expected that the latter, on passing 



REMARKABLE COMETS. 405 

the orbit of the comet, would intersect the fragmentary 
meteoroids supposed to follow it, as explained in the last 
chapter. According to the calculated orbit of the comet, it 
crossed the point of intersection in September, 1872, while 
the earth passes the same point on November 27th of each 
year. It was therefore predicted that a meteoric shower 
would be seen on the night of November 27th, the radiant 
point of which would be in the constellation Andromeda. 
This prediction was completely verified, but the meteors 
were so faint that though they succeeded each other quite 
rapidly, they might not have been noticed by a casual 
observer. They all radiated from the predicted point with 
such exactness that the eye could detect no deviation what- 
ever. 

We thus have a third case in which meteoric showers 
are associated with the orbit of a comet. In this case, how- 
ever, the comet has been completely dissipated, and proba- 
bly has disappeared forever from telescopic vision, though 
it may be expected that from time to time its invisible 
fragments will form meteors in the earth's atmosphere. 

§ 6. REMARKABLE COMETS. 

It is familiarly known that bright comets were in former 
years objects of great terror, being supposed to presage 
the fall of empires, the death of monarchs, the approach 
of earthquakes, wars, pestilence, and every other calamity 
which could afflict mankind. In showing the entire 
groundlessness of such fears, science has rendered one of its 
greatest benefits to mankind. 

In 1450, the comet known as Halley's, appearing 
when the Turks were making war on Christendom, caused 
such terror that Pope Calixtus ordered prayers to be 
offered in the churches for protection against it. This 
is supposed to be the origin of the popular myth that the 
Pope once issued a bull against the comet. 

The number of comets visible to the naked eye, so far as 



406 ASTRONOMY. 

recorded, lias generally ranged from 20 to 40 in a cen- 
tury. Only a small portion of these, however, nave been 
so bright as to excite universal notice. 

Comet of 1680. — One of the most remarkable of these 
brilliant comets is that of 1680. It inspired such terror 
that a medal, of which we present a figure, was struck 
upon the Continent of Europe to quiet apprehension. A 
free translation of the inscription is : ' ( The star threatens 
evil things; trust only ! God will turn them to good." 
"What makes this comet especially remarkable in history 
is that Newton calculated its orbit, and showed that it 
moved around the sun in a conic section, in obedience to 
the law of gravitation. 




Fig. 109. — medal of the great comet op 1680. 

Great Comet of 1811. — Fig. 110 shows its general ap- 
pearance. It has a period of over 3000 years, and its 
aphelion distance is about 40,000,000,000 miles. 

Great Comet of 1843. — One of the most brilliant com- 
ets which have appeared during the present century was 
that of February, 1843. It was visible in full daylight 
close to the sud. Considerable terror was caused in some 
quarters, lest it might presage the end of the world, 
which had been predicted for that year by Miller. At 
perihelion it passed nearer the sun than any other body 
has ever been known to pass, the least distance being only 
about one fifth of the sun's semi-diameter. With a very 
slight change of its original motion, it would have actually 
fallen into the sun. 



GREAT COMET OF 1858. 407 

Great Comet of 1858. — Another remarkable comet for 
the length of time it remained visible was that of 1858. 
It is frequently called after the name of Donati, its first 
discoverer. No comet visiting our neighborhood in 




Fig. 110 — great comet op 1811. 

recent times has afforded so favorable an opportunity for 
studying its physical constitution. Some of the results of 
the observations made upon it hove already been presented. 




Fig. 111.— donatt's comet of 1858. 



ENCKE'S COMET. 409 

Its greatest brilliancy occurred about the beginning of 
Octooer, when its tail was 40° in length and 10° in breadth 
at its outer end. 

Donati's comet had not long been observed when it 
was found that its orbit was decidedly elliptical. After it 
disappeared, the observations were all carefully investigated 
by two mathematicians, Dr. Yon Asten, of Germany, 
and Mr. G. "W. Hill, of this country. The latter found 
a period of 1950 years, which is probably within a half a 
century of the truth. It is probable, therefore, that this 
comet appeared about the first century before the Chris- 
tian era, and will return again about the year 3800. 



Eneke's Comet and the Resisting Medium. — Of telescopic 
comets, that which has been most investigated by astronomers is 
known as Encke's comet. Its period is between three and four 
years. Viewed with a telescope, it is not different in any respect 
from other telescopic comets, appearing simply as a mass of foggy 
light, somewhat brighter near one side. Under the most favorable 
circumstances, it is just visible to the naked eye. The circumstance 
which hasjent most interest to this comet is that the observations 
which have been made upon it seem to indicate that it is gradually 
approaching the sun. Encke attributed this change in its orbit to 
the existence in space of a resisting medium, so rare as to have no 
appreciable effect upon the motion of the planets, and to be felt 
only by bodies of extreme tenuity, like the telescopic comets. The 
approach of the comet to the sun is shown, not by direct obser- 
vation, but only by a gradual diminution of the period of revolu- 
tion. It will be many centuries before this period would be so far 
diminished that the comet would actually touch the sun. 

If the change in the period of this comet were actually due to 
the cause which Encke supposed, then other faint comets of the 
same kind ought to be subject to a similar influence. But the in- 
vestigations which have been made in recent times on these bodies 
show no deviation of the kind. It might, therefore, be concluded 
that the change in the period of Encke's comet must be due to 
some other cause. There is, however, one circumstance which 
leaves us in doubt. Encke's comet passes nearer the sun than any 
other comet of short period which has been observed with suffi- 
cient care to decide the question. It may, therefore, be supposed 
that the resisting medium, whatever it may be, is densest near the 
sun, and does not extend out far enough 'for the other comets to 
meet it. The question is one very difficult to settle. The fact is 
that all comets exhibit slight anomalies in their motions which pre- 
vent us from deducing conclusions from them with the same cer- 
tainty that we should from those of the planets. 



PART III. 

THE UNIVERSE AT LARGE. 



INTRODUCTION. 

In our studies of the heavenly bodies, we have hitherto 
been occupied almost entirely with those of the solar sys- 
tem. Although this system comprises the bodies which 
are most important to us, yet they form only an insignifi- 
cant part of creation. Besides the earth on which we 
dwell, only seven of the bodies of the solar system are 
plainly visible to the naked eye, whereas it is well known 
that 2000 stars or more can be seen on any clear night. 
We now have to describe the visible universe in its largest 
extent, and in doing so shall, in imagination, step over 
the bounds in which we have hitherto confined ourselves 
and fly through the immensity of space. 

The material universe, as revealed by modern telescopic 
investigation, consists principally of shining bodies, many 
millions in number, a few of the nearest and brightest of 
which are visible to the naked eye as stars. They extend 
out as far as the most powerful telescope can penetrate, 
and no one knows how much farther. Our sun is simply 
one of these stars, and does not, so far as we know, differ 
from its fellows in any essential characteristic. From the 
most careful estimates, it is rather less bright than the 
average of the nearer stars, and overpowers them by its 
brilliancy only because it is so much nearer to us. 

The distance of the stars from each other, and therefore 



412 ASTRONOMY. 

from tlie sun, is immensely greater than any of the dis- 
tances which we have hitherto had to consider in the solar 
system. Suppose, for instance, that a walker through 
the celestial spaces could start out from the sun, taking steps 
3000 miles long, or equal to the distance from Liverpool to 
New York, and making 120 steps a minute. This speed 
would carry him 'around the earth in about four seconds ; 
he would walk from the sun to the earth in four hours, and 
in five days he would reach the orbit of Neptune. Yet if 
he should start for the nearest star, he would not reach it 
in a hundred years. Long before he got there, the whole 
orbit of Neptune, supposing it a visible object, would have 
been reduced to a point, and have finally vanished from 
sight altogether. In fact, the nearest known star is about 
seven thousand times as far as the juanet Neptuy;^. If 
we suppose the orbit of this planet to be represented by a 
child's hoop, the nearest star would be three or four miles 
away. We have no reason to suppose that contiguous 
stars are, on the average, nearer than this, except in special 
cases where they are collected together in clusters. 

The total number of the stars is estimated by millions, 
and they are probably separated by these wide intervals. 
It follows that, in going from the sun to the nearest star, 
we would be simply taking one step in the universe. The 
most distant stars visible in great telescopes are probably 
several thousand times mere distant than the nearest one, 
and we do not know what may lie beyond. 

The point we wish principally to impress on the reader 
in this connection is that, although the stars and planets pre- 
sent to the naked eye so great a similarity in appearance, 
there is the greatest possible diversity in their distances 
and characters. The planets, though many millions of 
miles away, are comparatively near us, and form a little 
family by themselves, which is called the solar system. 
The fixed stars are at distances incomparably greater — the 
nearest star, as just stated, being thousands of times more 
distant than the farthest planet. The planets are, so far 



THE UNIVERSE AT LARGE. 



413 



as we can sec, worlds somewhat like this on which we live, 
while the stars are suns, generally larger and brighter than 
our own. Each star may, for aught we know, have plan- 
ets revolving around it, but their distance is so immense 
that the largest planets will remain invisible with the most 
powerful telescopes man can ever hope to construct. 

The classification of the heavenly bodies thus leads us to 
this curious conclusion. Our sun is one of the family of 
stars, the other members of which stud the heavens at 
night, or, in other words, the stars are suns like that which 
makes the day. The planets, though they look like stars, 
are not such, but bodies more like the earth on which 
we live. 

The great universe of stars, including the creation in its 
largest extent, is called the stellar system, or stellar 
universe. We have first to consider how it looks to the 
naked eye. 



CHAPTER I. 

THE CONSTELLATIONS. 
9 1. GENERAL ASPECT OP THE HEAVENS. 

When we view the heavens with the unassisted eye, the 
stars appear to be scattered nearly at random over the 
surface of the celestial vault. The only deviation from an 
entirely random distribution which can be noticed is a cer- 
tain grouping of the brighter ones into constellations. 
We notice also that a few are comparatively much brighter 
than the rest, and that there is every gradation of bril- 
liancy, from that of the brightest to those which are barely 
visible. We also notice at a glance that the fainter stars 
outnumber the bright ones ; so that if we divide the stars 
into classes according to their brilliancy, the fainter classes 
will be far the more numerous. 

The total number one can see will depend very largely 
upon the clearness of the atmosphere and the keenness of 
the eye. From the most careful estimates which have 
been made, it would appear that there are in the whole 
celestial sphere about 6000 stars visible to an ordinarily 
good eye. Of these, however, we can never see more than 
a fraction at any one time, because one half of the sphere is 
always of necessity below the horizon. If we could see a 
star in the horizon as easily as in the zenith, one half of the 
whole number, or 3000, would be visible on any clear night. 
But stars near the horizon are seen through so great a 
thickness of atmosphere as greatly to obscure their light ; 
consequently only the brightest ones can there be seen. As 



CLASSES OF STARS. 



415 



a result of this obscuration, it is not likely that more than 
2000 stars can ever be taken in at a single view by any 
ordinary eye. About 2000 other stars are so near the 
South Pole that they never rise in our latitudes. Hence 
out oi the 6000 supposed to be visible, only 4000 ever 
come within the range of our vision, unless we make a 
journey toward the equator. 

The Galaxy. — Another feature of the heavens, which is 
less striking than the stars, but has been noticed from 
the earliest times, is the Galaxy, or Milky Way. This 
object consists of a magnificent stream or belt of white 
milky light 10° or 15° in breadth, extending obliquely 
around the celestial sphere. During the spring months, it 
nearly coincides with our horizon in the early evening, 
but it can readily be seen at all other times of the year 
spanning the heavens like an arch. It is for a portion of 
its length split longitudinally into two parts, which remain 
separate through many degrees, and are finally united 
again. The student will obtain a better idea of it by 
actual examination than from any description. He will 
see that its irregularities of form and lustre are such that 
in some places it looks like a mass of brilliant clouds. In 
the southern hemisphere there are vacant spaces in it 
w^hich the navigators call coal-sacks. In one of these, 
5° by 18°, there is scarcely a single star visible to the 
naked eye (see Figs. 121 and 132). 

Lucid and Telescopic Stars. — When we view the 
heavens with a telescope, we find that there are innumer- 
able stars too small to be seen by the naked eye. We 
may therefore divide the stars, with respect to brightness, 
into two great classes. 

Lucid Stars are those which are visible without a tele- 
scope. 

Telescopic Stars are those which are not so visible. 

When Galileo first directed his telescope to the heav- 
ens, about the year 1610, he perceived that the Milky 
Way was composed of stars too faint to be individually 



416 ASTRONOMY. 

seen by the unaided eye. We thus have the interesting 
fact that although telescopic stars cannot be seen one by 
one, yet in the region of the Milky Way they are so numer- 
ous that they shine in masses like brilliant clouds. Huy- 
ghens in 1656 resolved a large portion of the Galaxy into 
stars, and concluded that it was composed entirely of them. 
Kepler considered it to be a vast ring of stars surround- 
ing the solar system, and remarked that the sun must be 
situated near the centre of the ring. This view agrees 
very well with the one now received, only that the stars 
which form the Milky Way, instead of lying around the 
solar system, are at a distance so vast as to elude all our 
powers of calculation. 

Such are in brief the more salient phenomena which 
are presented to an observer of the starry heavens. We 
shall now consider how these phenomena have been clas- 
sified by an arrangement of the stars according to theii 
brilliancy and their situation. 

§ 2. MAGNITUDES OP THE STARS. 

In ancient times, the stars were arbitrarily classified into 
six orders of magnitude. The fourteen brightest visible in 
our latitude were designated as of the first magnitude, while 
those which were barely visible to the naked eye were said 
to be of the sixth magnitude. This classification, it will 
be noticed, is entirely arbitrary, since there are no two 
stars which are absolutely of the same brightness, while if 
all the stars were arranged in the order of their actual 
brilliancy, we should find a regular gradation from the 
brightest to the faintest, no two being precisely the same. 
Therefore the brightest star of any one magnitude is 
about of the same brilliancy with the faintest one of the 
next higher magnitude. It depends upon the judgment 
of the observer to what magnitude a given star shall be 
assigned ; so that we cannot expect an agreement on this 
point. The most recent and careful division into magni- 



MAGNITUDES, OF STARS. 



417 



tudes has been made by Heis, of Germany, whose results 
with respect to numbers are as follows. Between the 
^N"orth Pole and 35° south declination, there are : 

14 stars of the first magnitude, 

, 48 " " second " 

152 " " third " 

313 " " fourth " 

854 " " fifth 

3974 " . " sixth " 



5355 of the first six magnitudes. 

Of these, however, nearly 2000 of the sixth magnitude 
are so faint that they can be seen only by an eye of extra- 
ordinary keenness. 



In order to secure a more accurate classification and expression of 
brightness, Heis and others have divided each magnitude into 
three orders or sub-magnitudes, making eighteen orders in all 
visible to the naked eye. When a star was considered as falling be- 
tween two magnitudes, both figures were written, putting the mag- 
nitude to which the star most nearly approached first. For in- 
stance, the faintest stars of the fourth magnitude were called 4-5. 
The next order below this would be the brightest of the fifth 
magnitude ; these were called 5*4. The stars of the average fifth 
magnitude were called 5 simply. The fainter ones were called 5 '6, 
and so on. This notation is still used by some astronomers, but 
those who aim at greater order and precision express the magni- 
tudes in tenths. For instance, the bi ightest stars of the fifth magni- 
tude they would call 4-6, those one tenth fainter 4-7, and soon 
until they reached the average of the fifth magnitude, which 
would be 5-0. The division into tenths of magnitudes is as mi- 
nute a one as the practised eye is able to make. 

This method of designating the brilliancy of a star on a scale of 
magnitudes is not at all accurate. Several attempts have been 
made in recent times to obtain more accurate determinations, by 
measuring the light of the stars. An instrument with which this 
can be done is called a photometer. The results obtained with the 
photometer have been used to correct the scale of magnitudes 
and make it give a more accurate expression for the light of the 
stars. The study of such measures shows that, for the most part, 
the brightness of the stars increases in geometrical progression as 
the magnitudes vary in arithmetical progression. The stars of one 
magnitude are generally about %\ times as bright as those of the 
magnitude next below it. Therefore if we take the light of a star 



418 ASTRONOMY. 

of the sixth magnitude, which is just visible to the naked eye, as 
unity, we shall have the following scale : 

Magnitude 6th, brightness 1 
5th, " 2£ 

" 4th, " 6i 

" 3d, " 16 nearly 

" 2d, " 40 

" 1st, " 100 

Therefore, according to these estimates, an average star of the 
first magnitude is about 100 times as bright as one of the sixth. 
There is, however, a deviation from this scale in the case of the 
brighter magnitudes, an average star of the second magnitude 
being perhaps three times as bright as one of the third, and most 
of the stars of the first magnitude brighter than those of the second 
in a yet larger ratio. Indeed, the first magnitude stars differ so 
greatly in brightness that we cannot say how bright a standard 
star of that magnitude really is. Sirim, for instance, is probably 
500 times as bright as a sixth magnitude star. 

The logarithm of 2£ being very nearly 0*40, we can readily find 
how many stars of any one magnitude are necessary to make one of 
the higher magnitude by multiplying the difference of the magni- 
tude by 0*40, and taking the number corresponding to this logarithm. 

This scale will enable us to calculate in a rough way the magni- 
tude of the smallest stars which can be seen with a telescope of given 
aperture. The quantity of light which a telescope admits is directly 
as the square of its aperture. The amount of light emitted by the 
faintest star visible in it is therefore inversely as this square. If we 
increase the aperture 50 per cent, we increase the seeing power of 
our telescope about one magnitude. More exactly, the ratio of in- 
crease of aperture is V 2$, or 1 • 58. The pupil of the eye is probably 
equivalent to a telescope of about J of an inch in aperture ; that 
is, in a telescope of this size the faintest visible star would be about 
of the sixth magnitude. To find the exact magnitude of the 
faintest star visible with a larger telescope, we recall that the 
quantity of light received by the objective is proportional to the 
square of the aperture. As just shown, every time we multiply the 
square of the aperture by 21, or the aperture itself by the square 
root of this quantity, we add one magnitude to the power of our 
telescope. Therefore, if we call a the aperture of a telescope 
which would just show a star one magnitude brighter than the 
first (or mag. 0), the aperture necessary to show a star of magnitude 
m will be found by multiplying a by 1*58 m times — that is, it will 
be 1.58 ul a a . So, calling a this aperture, we have : 



a = l-58 m <io = a V2-5 m . 

Taking the logarithms of both sides of the equation, and using ap- 
proximate round numbers which are exact enough for this purpose : 

log. a = m log. 1-58 + log. a Q — — log. 2-5 ■+- log. a = — + log.a . 

A 



KAMES OF THE STABS. 419 

Now, as just found, when m = 6, a = ln -25 = 6*4 millimetres. 
With these values of a and m we find • 

log. a = — 1*802 in fractions of an inch. 

= — 0-397 in fractions of a millimetre. 

Hence, when the magnitude is given, and we wish to find the aperture : 

log. a = — — 1-802 [will give aperture in inches.] 
5 

log. a = — — 0-397 [will give aperture in millimetres. j 
5 

If the aperture is given, and we require the limiting magnitude . 

m = 5 log. a + 9 «0 [if a is in inches.] 

m = 5 log. a -+- 2-0 [if a is in millimetres.] 

The magnitudes for different apertures is shown in the following 
table : 



Aperture. 


Minimum 
Visibile. 


Aperture. 


Minimum 
Visibile. 


Inches. 


Magnitude. 


Inches. 


Magnitude. 


1-0 


9-0 


6-5 


131 


1-5 


9-9 


70 


13-3 


2-a 


10-5 


8-0 


13-5 


2-5 


11-0 


90 


13-8 


30 


11-4 


100 


140 


3-5 


11-7 


110 


14-2 


4-0 


12-0 


120 


14-4 


4-5 


123 


15-0 


14-9 


50 


12-5 


18-0 


15-3 


5-5 


127 


26-0 


16-1 


6-0 


12-9 


34-0 


16-6 



§ 3. THE CONSTELLATIONS AND NAMES OF THE 

STARS. 

The earliest astronomers divided the stars into groups, 
called constellations, and gave special proper names both 
to these groups and to many of the more conspicuous 
stars. We have no record of the process by which this 
was done, or of the considerations which led to it. It was 
long before the commencement of history, as we may in- 
fer from different allusions to the stars and constellations 
in the book of Job, which is supposed to be among the 



420 ASTRONOMY. 

most ancient writings now extant. We have evidence 
that more than 3000 years before the commencement of 
the Christian chronology the star Sirius, the brightest in 
the heavens, was known to the Egyptians under the name 
of Sothis. Arcturus is mentioned by Job himself. The 
seven stars of the Great Bear, so conspicuous in our north- 
ern sky, were known under that name to Homer and He 
stod, as well as the group of the Pleiades, or Seven Stars, 
and the constellation of Orion. Indeed, it would seem 
that all the earlier civilized nations, Egyptians, Chinese, 
Greeks, and Hindoos, had some arbitrary division of the 
surface of the heavens into irregular, and often fantastic 
shapes, which were distinguished by names. 

In early times, the names of heroes and animals were 
given to the constellations, and these designations have 
come down to the present day. Each object was sap- 
posed to be painted on the surface of the heavens, and the 
stars were designated by their position upon some portion 
of the object. The ancient and mediaeval astronomers 
would speak of ' ' the bright star in the left foot of 
Orion, " " the eye of the Bull, " " the heart of the Lion, ' ' 
" the head of Perseus," etc. These figures are still re- 
tained upon some star-charts, and are useful where it is 
desired to compare the older descriptions of the constella- 
tions with our modern maps. Otherwise they have ceased 
to serve any purpose, and are not generally found on maps 
designed for astronomical uses. 

The Arabians, who used this clumsy way of designating 
stars, gave special names to a large number of the brighter 
ones. Some of these names are in common use at the 
present time, as Aldebaran, Fomalhaut, etc. A few other 
names of bright stars have come down from prehistoric 
times, that of Arcturus for instance : they are, how- 
ever, gradually falling out of use, a system of nomencla- 
ture introduced in modern times having been substituted. 

In 1654, Bayer, of Germany, mapped down the constel- 
lations upon charts, designating the brighter stars of each 



NAMING THE STARS. 421 

constellation by the letters of the Greek alphabet. When 
this alphabet was exhausted, he introduced the letters of 
the Eoman alphabet. In general, the brightest star was 
-designated by the first letter of the alphabet a, the next 
by the following letter /3, etc. Although this is sometimes 
.supposed to have been his rule, the Greek letter affords 
only an imperfect clue to the average magnitude of a star. 
In a great many of the constellations there are deviations 
from the order, the brightest star being (3 ; but where stars 
differ by an entire magnitude or more, the fainter ones 
nearly always follow the brighter ones in alphabetical order. 

On this system, a star is designated by a certain Greek 
letter, followed by the genitive of the Latin name of the 
constellation to which it belongs. For example, a Canis 
Majoris, or, in English, a of the Great Dog, is the desig- 
nation of Sirius, the brightest star in the heavens. The 
.seven stars of the Great Bear are called a Ursce Majoris, 
*j3 Ursce Majoris, etc. Arcturus is a B otitis. The 
reader will here see a resemblance to our way of designat- 
ing individuals by a Christian name followed by the family 
name. The Greek letters furnish the Christian names of 
-the separate stars, while the name of the constellation is 
that of the family. As there are only fifty letters in the 
"two alphabets used by Bayer, it will be seen that only the 
fifty brightest stars in each constellation could be desig- 
nated by this method. In most of the constellations the 
number thus chosen is much less than fifty. 

When by the aid of the telescope many more stars than 
these were laid down, some other method of denoting 
Tthem became necessary. Flamsteed, who observed be- 
fore and after 1700, prepared an extensive catalogue of 
.stars, in which those of each constellation were designated 
by numbers in the order of right ascension. These num- 
bers were entirely independent of the designations of 
Bayer — that is, he did not omit the Bayer stars from 
his system of numbers, but numbered them as if they had 
no Greek letter. Hence those stars to which Bayer ap- 



422 ASTRONOMY. 

plied letters have two designations, the letter and the 
number. 

Flamsteed's numbers do not go much above 100 for 
any one constellation — Taurus, the richest, having 139. 
When we consider the more numerous minute stars, no 
systematic method of naming them is possible. The star 
can be designated only by its position in the heavens, or 
the number which it bears in some well-known catalogue. 

§ 4. DESCRIPTION OF THE CONSTELLATIONS. 

The aspect of the starry heavens is so pleasing that 
nearly- every intelligent person desires to possess some 
knowledge of the names and forms of the principal con- 
stellations. We therefore present a brief description of 
the more striking ones, illustrated by figures, so that the 
reader may be able to recognize them when he sees them 
on a clear night. 

We begin with the constellations near the pole, because 
they can be seen on any clear night, while the southern 
ones can, for the most part, only be seen during certain 
seasons, or at certain hours of the night. The accompany- 
ing figure shows all the stars within 50° of the pole down 
to the fourth magnitude inclusive. The Roman numerals 
around the margin show the meridians of right ascension, 
one for every hour. In order to have the map represent 
the northern constellations exactly as they are, it must be 
held so that the hour of sidereal time at which the observer 
is looking at the heavens shall be at the top of the map 
Supposing the observer to look at nine o'clock in the even- 
ing, the months around the margin of the map show the 
regions near the zenith. He has therefore only to hold the 
map with the month upward and face the north, when he 
will have the northern heavens as they appear, except 
that the stars near the bottom of the map will be cut on 5 
by the horizon. 

The first constellation to be looked for is Ursa Major y 



THE CONSTELLATIONS. 



423 



the Great Bear, familiarly known as " the Dipper. " The 
two extreme stars in this constellation point toward the 
pole-star, as already explained in the opening chapter. 

Ursa Minor, sometimes called " the Little Dipper," is 
the constellation to which the pole-star belongs. Abouf 




Fig. 112. — map of the northern constellations. 

15° from the pole, in right ascension XV. hours, is a stai 
of the second magnitude, fi Ui^sai Minor is, about as bright 
as the pole-star. A curved row of three small stars lies 
between these two bright ones, and forms the handle of 
the supposed dipper. 



424 ASTRONOMY. 

Cassiopeia, or " the Lady in the Chair," is near hour I 
of right ascension, on the opposite side of the pole-star 
from Ursa Major, and at nearly the same distance. 
The six brighter stars are supposed to bear a rude resem- 
blance to a chair. In mythology, Cassiopeia was the queen 
of Cepheus, and in the mythological representation of the 
constellation she is seated in the chair from which she is 
issuing her edicts. 

In hour III of right ascension is situated the constella- 
tion Perseus, about 10° further from the pole than Cas- 
siopeia. The Milky Way passes through these two con- 
stellations. 

Draco, the Dragon, is formed principally of a long 
row of stars lying between Ursa Major and Ursa Minor. 
The head of the monster is formed of the northernmost 
three of four bright stars arranged at the corners of a 
lozenge between XYII and XYIII hours of right ascen- 
sion. 

Cepheus is on the opposite side of Cassiopeia from 
Perseus , lying in the Milky Way, about XXII hours of 
right ascension. It is not a brilliant constellation. 

Other constellations near the pole are Camelopardalis, 
Lynx, and Lacerta (the Lizard), but they contain only 
small stars. 

In describing the southern constellations, we shall take 
four separate positions of the starry sphere corresponding 
respectively to VI hours, XII hours, XYIII hours, 
and hours of sidereal time or right ascension. These 
hours of course occur every day, but not always at con- 
venient times, because they vary with the time of the 
year, as explained in Chapter I. , Part I. 

We shall first suppose the observer to view the heavens 
at YI hours of sidereal time, which occurs on Decem- 
ber 21st about midnight, January 1st about 11.30 p.m., 
February 1st about 9.30 p.m., March 1st about 7.30 
p.m., and so on through the year, two hours earlier every 
month. In this position of the sphere, the Milky Way 



THE CONSTELLATIONS. 



425 



spans the heavens like an arch, resting on the horizon be- 
tween north and north-west on one side, and between 
south and south-east on the other. We shall first describe 
the constellations which lie in its course, beginning at the 
north. Cepheus is near the north-west horizon, and above 
it is Cassiopeia, distinctly visible at an altitude nearly 
equal to that of the pole. Next is Perseus ■, just north- 
west of the zenith. Above Perseus lies Auriga, the 
Charioteer, which may be recognized by a bright star of 
the first magnitude called Capella (the Goat), now quite 
near the zenith. Auriga is represented as holding a 
goat in his arms, in the body of which the star is situated. 
About 10° east of Capella is the star Aurigw of the 
second magnitude. 

Going further south, the Milky Way next passes between 
Taurus and Gemini. 

Taurus, the Bull, may be recognized by the Pleiades, 
or " Seven Stars." Really there are only six stars in the 
group clearly visible to ordi- 
nary eyes, and any eye strong 
enough to see seven will prob- 
ably see four others, or eleven 
in all. This group forms an 
interesting object of study 
with a small telescope, as sixty 
or eighty stars can then readily 
be seen. We therefore pre- 
sent a telescopic view of it, 
the six large stars being those 
visible to any ordinary eye, 
the five next in size those 
which can be seen by a re- 
markably good eye, and the 
others those which require a telescope. East of the Pleia- 
des is the bright red star Aldebaran, or " the Eye of 
the Bull." It lies in a group called the Hyades^ ar- 
ranged in the form of the letter V, and forming the face 




Fig. 113.— view of the Plei- 
ades AS SEEN TN AN INVERT- 
ING TELESCOPE 



426 



ASTRONOMY. 



of the Bull. In the middle of one of the legs of the V 
will be seen a beautiful pair of stars of the fourth magni- 
tude very close together. They are called 6 Tauri. 

Gemini, the Twins, lie east of the Milky Way, and 
may be recognized by the bright stars Castor and JPollux, 
which lie 20° or 30° south-east or south of the zenith. 




Fig. 114. — the constellation orion. 



They are about 5° apart, and Pollux, the southernmost 
one, is a little brighter than Castor. 

Orion, the most brilliant constellation in the heavens, 
is very near the meridian, lying south-east of Taurus and 
south-west of Gemini. It may be readily recognized by 
the figure which we give. Four of its bright stars form 



THE CONSTELLATIONS. 427 

the corners of a rectangle about 15° long from north 
to south, and 5° wide. In the middle of it is a row of 
three bright stars of the second magnitude, which no one 
can fail to recognize. Below this is another row of three 
smaller ones. The middle star of this last row is called 
6 Orionis, and is situated in the midst of the great nebula 
of Orion, one of the most remarkable telescopic objects in 
the heavens. Indeed, to the naked eye this star has a 
nebulous hazy appearance. The two stars of the first 
magnitude are a Orionis, or Betelguese, which is the high- 
est, and may be recognized by its red color, and Rigel, 
or /3 Orionis, a sparkling white star lower down and a 
little to the west. The former is in the shoulder of the 
figure, the latter in the foot. A little north-west of 
Betelguese are three small stars, which form the head. 
The row of stars on the west form his arm and club, the 
latter being raised as if to strike at Taurus, the Bull, on 
the west. 

Canis Minor, the Little Dog, lies across the Milky 
Way from Orion, and may be recognized by the bright 
star Procyon of the first magnitude. The three stars 
Pollux, Procyon, and Betelguese form a right-angled tri- 
angle, the right angle being at Procyon. 

Canis Major, the Great Dog, lies south-east of Orion, 
and is easily recognized by Sirius, the brightest fixed star 
in the heavens. A number of bright stars south and 
south-east of Sirius belong to this constellation, making 
it one of great brilliancy. 

Argo Navis, the ship Argo, lies near the south horizon, 
partly above it and partly below it. Its brightest star is 
Oanopus, which, next to Sirius, is the brightest star in 
the heavens. Being in 53° of south declination, it never 
rises to an observer within 53° of the North Pole — that is, 
north of 37° of north latitude. In our country it is visi- 
ble only in the Southern States, and even there only 
between six and seven hours of sidereal time. 

We next trace out the zodiacal constellations, which are 



£28 ASTRONOMY. 

of interest because it is through them that the sun passes 
in its apparent annual course. We shall commence in 
the west and go toward the east, in the order of right 
ascension. 

Aries, the Ham, is in the west, about one third of the 
way from the horizon to the zenith. It may be recognized 
by three stars of the second, third, and fourth magni- 
tudes, respectively, forming an obtuse-angled triangle. 
The brightest star is the highest. Next toward the east 
is Taurus, the Bull, which brings us nearly to the meri- 
dian, and east of the meridian lies Gemini, the Twins, both 
of which constellations have just been described. 




Fig. 115 —the constellation leo, the lion. 

Cancer, the Crab, lies east of Gemini, but contains no 
Dright star. The most noteworthy object in this constel- 
lation is Prwsepe, a group of telescopic stars, which ap- 
pears to the naked eye like a spot of milky light. To see 
it well, the night must be clear and the moon not in the 
neighborhood. 

Leo, the Lion, is from one to two hours above the 
oa stern horizon. Its brightest star is Regulus, one third 
of the way from the eastern horizon to the zenith, and 
between the first and second magnitudes. Five or six 
stars north of it in a curved line are in the form of a 



THE CONSTELLATIONS. 429 

sickle, of which Begulus is the handle. As the Lion was 
drawn among the old constellations, Begulus formed his 
heart, and was therefore called Cor Zeonis. The sickle 
forms his head, and his body and tail extend toward the 
horizon. The tail ends near the star Denebola, which is 
quite near the horizon. 

Leo Minor lies to the north of Leo, and Sextans, the 
Sextant, south of it, but neither contains any bright stars. 

Eridanus, the River Po, south-west of Orion / Lepus, 
the Hare, south of Orion and west of Canis Major / 
Columba, the Dove, south of Lepus, are constellations in 
the south and south-west, which, however, have no strik- 
ing features. 

The constellations we have described are those seen at 
six hours of sidereal time. If the sky is observed at some 
other hour near this, we may find the sidereal time by the 
rule given in Chapter I., § 5, p. 30, and allow for the di- 
urnal motion during the interval. 

Appearance of the Constellations, at 1 2 Hours Sidereal 
Time. — This hour occurs on April 1st at 11.30 p.m., on 
May 1st at 9.30 p.m., and on June 1st at 7.30 p.m. 

At this hour, Ursa Major is near the zenith, and Cassi- 
opeia near or below the north horizon. The Milky Way 
is too near the horizon to be visible. Orion has set in 
the west, and there is no very conspicuous constellation 
in the south. Castor and Pollux are high up in the 
north-west, and Procyon is about an hour and a half 
above the horizon, a little to the south of west. All the 
constellations in the west and north-west have been previ- 
ously described, Leo being a little west of the meridian. 
Three zodiacal constellations have, however, risen, which 
we shall describe. 

Virgo, the Yirgin, has a single bright star, Spica, 
about as bright as Begulus, now about one hour east of 
the meridian, and but little more than half way from the 
zenith to the horizon. 

Libra, the Balance, is south-east from Virgo, but lias 
no conspicuous stars. 



430 ASTRONOMY. 

Scorpius, the Scorpion, is just rising in the south-east, 
but is not yet high enough to be well seen. 

Hydra is a very long constellation extending from 
Canis Minor in a south-east direction to the south hori- 
zon. Its brightest star is a Hydra, of the second magni- 
tude, 25° below Regulus. 

Corvus, the Crow, is south of Virgo, and may be recog- 
nized by four or five stars of the second or third magni- 
tude, 15° south-west from Spica. 

Next, looking north of the zodiacal constellations, we 
see : 

Coma Berenices, the Hair of Berenice, now exactly on 
the meridian, and about 10° south of the zenith. It is a 
close irregular cluster of very small stars, unlike any thing 
else in the heavens. In ancient mythology, Berenice had 
vowed her hair to Yenus, but Jupiter carried it away from 
the temple in which it was deposited, and made it into a 
constellation. 

Bootes, the Bear-Keeper, is a large constellation east of 
Coma Berenices. It is marked by Arcturus, a bright but 
somewhat red star of the first magnitude, about 20° east 

of the zenith. Bootes is repre- 
sented as holding two dogs in a 
leash. These dogs are called 
Canes Venatici, and are at the 
time supposed exactly in our ze- 
nith chasing Ursa Major around 
the pole. 

Corona Borealis, the North- 
ern Crown, lies next east of 
Fig. 116.-COBOKA bore- Bootes in the north-east. It is 

a small but extremely beautiful 
constellation. Its principal stars are arranged in the form 
of a semicircular chaplet or crown. 

Appearance of the Constellations at 18 Hours of Side- 
real Time. — This hour occurs on July 1st at 11.30 p.m., 
on August 1st at 9.30 p.m., and on September 1st at 7.30 

P.M. 




THE CONSTELLATIONS. 



431 



In this position, the Milky Way seems once more to 
span the heavens like an arch, resting on the horizon in 
the north-east and south-west. But we do not see the 
same parts of it which were visible in the first position at 
six hours of right ascension. Cassiopeia is now in the 
north-east and Ursa Major has passed over to the 
west. 

Arcturus is two or tin ee hours above the western hori- 
zon. "We shall commence, as in the first position of the 
sphere, by describing the constellations which lie along on 
the Milky Way, starting from Cassiopeia. Above Cassi- 
opeia we have Cepheus, and then Lacerta, neither of 
which contains any striking stars. 

Cygnus, the Swan, may be recognized by four or five 
stars forming a cross directly in the centre of the Milky 
Way, and a short distance north-east from the zenith. 
The brightest of these stars, a Cygni, forms the northern 
end of the cross, and is nearly of the first magnitude. 

Lyra, the Harp, is a beautiful constellation south-west 
of Cygnus, and nearly in the zenith. It contains the 
brilliant star Vega, or a 
Lyrae, of the first mag- 
nitude, and of a bluish 
white color. South of 
Vega are four stars of 
the fourth magnitude, 
forming an oblique par- 
allelogram, by which the 
constellation can be read- 
ily recognized. East of 
Vega, and about as far 
from it as the nearest 
star of the parallelogram, is s Lyrm, a very interesting 
object, because it is really composed of two stars of the 
fourth magnitude, which can be seen separately by a very 
keen eye. The power to see this star double is one of the 
best tests of the acuteness of one's vision (see Fig. 122). 




Fig. 117. — lyra, the harp. 



432 



ASTRONOMY. 




Fig. 118 

nus, and sagitta 



Aquila, the Eagle, is the next striking constellation in 
the Milky Way. It is two hours east of the meridian, 

and about midway between the 
zenith and horizon. It is readily 
recognized by the bright star 
Altair or a Aquilce, situated be- 
tween two smaller ones, the one 
of the third and the other of the 
fourth .magnitude. The row of 
three stars lies in the centre of 
the Milky Way. 

Sagitta, the Arrow, is a very 
small constellation, formed of 
three stars immediately north of 

AQUILA, DELPHI- AqUlld. 

Delphinus, the Dolphin, is a 
striking little constellation north-east of Aquila, recog- 
nized by four stars in the form of a lozenge. It is famil- 
iarly called " Job's Coffin." 

In this position of the celestial sphere three new zodia- 
cal constellations have arisen. 

Scorpius, the Scorpion, 
already mentioned, now two 
hours west of the meridian, 
and about 30° above the 
horizon, is quite a brilliant 
constellation. It contains An- 
tares, or a Scorpii, a red- 
dish star of nearly the first 
magnitude, and a long curv- 
ed row of stars west of it. 

Sagittarius, the Archer, 
comprises a large collection 
of second magnitude stars in 
and near the Milky Way, 
and now very near the meridian, 
form the arrow of the archer. 




SCOR- 



The westernmost stars 



THE CONSTELLATIONS. 



433 



Capricornus^ the Goat, is now in the south-east, but 
contains no bright stars. Aquarhis, the Water-bearer, 
which has just risen, and Pisces, the Fishes, which have 
partly risen, contain no striking objects. 

Ophiuchus, the Serpent-bearer, is a very large constel- 
lation north of Scorpius and west of the Milky Way. 
Ophiuchus holds in his hands an immense serpent, lying 
with its tail in an opening of the Milky Way, south-west 
of Aquila, while its head and body are formed of a col- 
lection of stars of the third and fourth magnitudes, ex- 
tending north of Scorpius nearly to Bootes. 

Hercules is a very 
large constellation 
between Corona 
Borealis and Lyra. 
It is now in the 
zenith, but contains 
no bright stars. It 
has, however, a 
number of interest- 
ing telescopic ob- 
jects, among them 
the great cluster of 
Hercules, barely 
visible to the naked 
eye, but containing 

an almost countless mass of stars. The head of Draco, 
already described, is just north of Hercules. 

Constellations Visible at Hours of Sidereal Time. — 
This time will occur on October 1st at 11.30 p.m., on 
November 1st at 9.30 p.m., on December 1st at 7.30 p.m. 
and on January 1st at 5.30 p.m. 

In this position, the Milky Way appears resting in the 
-east and west horizons, but in the zenith it is inclined 
over toward the north. All the constellations, either in 
or north of its course, are among those already described. 
We shall therefore consider only those in the south. 




FlG. 120.— SAGITTARIUS, THE ARCHER. 



434 ASTRONOMY. 

Pegasus, the Flying Horse, is distinguished by four 
stars of the second magnitude, which form a large square 
about 15° on each side, called the square of Pegasus. The 
eastern side of this square is almost exactly on the meri- 
dian. 

Andromeda is distinguished by a row of three or four 
bright stars, extending from the north-east corner of 
Pegasus, in the direction of Perseus. 

Cetus, the Whale, is a large constellation in the south 
and south-east. Its brightest star is fi Ceti, standing 
alone, 30° above the horizon, and a little east of the 
meridian. 

Piscis Australis, the Southern Fish, lies further west 
than Cetus. It has the bright star Fomalhaut, about 
15 c above the horizon, and an hour west of the meridian. 

§ 5. IHTMBERING AND CATALOGUING- THE STARS. 

As telescopic power is increased, we still find stars of 
fainter and fainter light. But the number cannot go on 
increasing forever in the same ratio as with the brighter 
magnitudes, because, if it did, the whole sky would be a 
blaze of starlight. 

If telescopes with powers far exceeding our present ones 
were made, they would no doubt show new stars of the 
20th and 21st magnitudes. But it is highly probable that 
the number of such successive orders of stars would not 
increase in the same ratio as is observed in the 8th, 9th, 
and 10th magnitudes, for example. The enormous labor 
of estimating the number of stars of such classes will long 
prevent the accumulation of statistics on this question ; 
but this much is certain, that in special regions of the sky. 
which have been searchingly examined by various tele- 
scopes of successively increasing apertures, the number of 
new stars found is by no means in proportion to the 
increased instrumental power. Thus, in the central por- 
tions of the nebula of Orion, only some half dozen stars 



CATALOGUING THE STARS. 



435 



lave been found with the Washington 26-inch refractor 
which were not seen with the Cambridge 15-inch, 
although the visible magnitude has been extended from 
15 m -l to 16 m, 3. If this is found to be true elsewhere, the 
conclusion may be that, after all, the stellar system can be 
experimentally shown to be of finite extent, and to contain 
only a finite number of stars. 



We have already stated that in the whole sky an eye of average 
power will see about 6000 stars. With a telescope this number is 
greatly increased, and the most powerful telescopes of modern times 
will probably show more than 60,000,000 stars. As no trustworthy 
estimate has ever been made, there is great uncertainty upon this 
point, and the actual number may range anywhere between 
40,000,000 and 100,000,000. Of this number, not one out of twenty 
has ever been catalogued at all. 

The gradual increase in the number of stars laid down in various 
of the older catalogues is exhibited in the following table from 
•Chambers's Descriptive Astronomy : 



Constella- 
tion. 


Ptolemy. 
B.C. 130. 


Tycho 

Brahe. 

a.d. 1570. 


Hevelius. Flamsteed. 
A.D. 1660. | A.D. 1690. 

j 


Bode. 
A.D. 1800. 


Aries 

Ursa Major.. 

Bootes 

Leo 

Tlrgo 

Taurus 

Orion 


18 

35 
23 
35 
32 
44 
38 


21 

56 
28 
40 
39 
43 
62 


27 
73 
52 
50 
50 
51 
62 


66 
87 
54 
95 
. 110 
141 
78 


148 
338 
319 
337 
411 
394 
304 



The most famous and extensive series of star observations are 
noticed below. 

The uranometries of Bayer, Flamsteed, Argelander, Heis, and 
Gould give the lucid stars of one or both hemispheres laid down 
on maps. They are supplemented by the star catalogues of other 
observers, of which a great number has been published. These last 
were undertaken mainly for the determination of star positions, but 
they usually give as an auxiliary datum the magnitude of the star 
observed. When they are carried so far as to cover the heavens, 
they will afford valuable data as to the distribution of stars 
throughout the sky. 

The most complete catalogue of stars yet constructed is the 
Durchmusterimg des Nordlichen Gestirnten Himmeh, the joint work 
of Argelander and his assistants, Kruger and Schonfeld. It 
embraces all the stars of the first nine magnitudes from the North 



436 ASTRONOMY 

Pole to 2° of south declination. This work was begun in 1852, and 
at its completion a catalogue of the approximate places of no less 
than 314,926 stars, with a series of star-maps, giving the aspect of 
the northern heavens for 1855, was published for the use of astrono- 
mers. Arge lander's original plan was to carry this Durchmusterung 
as far as 23° south, so that every star visible in a small comet-seeker 
of 2f inches aperture should be registered. His original plan was 
abandoned, but his former assistant and present successor at the 
observatory of Bonn, Dr. Schonfeld, is now engaged in executing 
this important work. 

The Catalogue of Stars of the British Association for the Ad- 
vancement of Science contains 8377 stars in both hemispheres, and 
gives all the stars visible to the eye. It is well adapted to 
learn the unequal distribution of the lucid stars over the celestial 
sphere. The table on the opposite page is formed from its data. 

From this table it follows that the southern sky has many more 
stars of the first seven magnitudes than the northern, and that the 
zones immediately north and south of the Equator, although greater 
in surface than any others of the same width in declination, are 
absolutely poorer in such stars. 

The meaning of the table will be much better understood by con- 
sulting the graphical representation of it on page 438, by Proctor. 
On this chart are laid down all the stars of the British Association 
Catalogue (a dot for each star), and beside these the Milky Way is 
represented. The relative richness of the various zones can be at 
once seen, and perhaps the scale of the map will allow the student 
to trace also the zone of brighter stars (lst-3d magnitude), which is 
inclined to that of the Milky Way by a few degrees, and is approx- 
imately a great circle of the sphere. 

The distribution and number of the brighter stars (1st- 7th mag- 
nitude) can be well understood from this chart. 

In Argelander's Durchmusterung of the stars of the northern, 
heavens, there are recorded as belonging to the northern hemi- 
sphere : 

10 stars between the 1-0 magnitude and the 1-9 magnitude. 

37 " " 2-0 " " 2-9 

128 " " 3-0 " «' 3-9 

310 " " 40 " " 4-9 

1,016 " " 5-0 " " 5-9 

4,328 " " 6-0 " " 6-9 

13,593 " " 7-0 " " 7-9 

57,960 " " 8-0 " " 8-9 

237,544 " " 9-0 " " 9-5 

In all 314,926 stars from the first to the 9-5 magnitudes are enu- 
merated in the northern sky, so that there are about 600,000 in the 
whole heavens. 

We may readily compute the amount of light received by the 
earth on a clear but moonless night fiom these stars. Let us assume 



i 

B 


DISTRIBUTION OF STARS. 

WMMMMMI-il-iPHil-'MMI-* O 

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437 



BRIGHTNESS OF THE STARS. 



439 



that the brightness of an average star of the first magnitude is 
about 0-5 of that of a Lyrce. A star of the 2d magnitude will shine 
with a light expressed by 0-5 x 0-4=0-20, and so on. 

The total brightness of 10 1st magnitude stars is 5-0 



M 


M 


37 2d 


7-4 


t* 


M 


122 3d 


10-1 


*t 


M 


310 4th 


9-9 


U 


" 


1,016 5th 


130 


U 


(( 


4,322 6th 


221 


a 


it 


13,593 7th 


27-8 


<t 


M 


57,960 8th 


47-4 



Sum =142-7 

It thus appears that from the stars to the 8th magnitude, inclu- 
sive, we receive 143 times as much light as from a Lyrce. a Lyrce 
has been determined by Zollner to be about 44,000,000,000 times 
fainter than the sun, so that the proportion of starlight to sunlight 
can be computed. It also appears that the stars of magnitudes too 
high to allow them to be individually visible to the naked eye are 
yet ao numerous as to affect the general brightness of the sky more 
tha» the so-called lucid stars (lst-6th magnitude). 



CHAPTER II. 

VARIABLE AND TEMPORARY STARS. 
§ 1. STARS REGULARLY VARIABLE. 

All stars do not shine with a constant light. Since 
the middle of the seventeenth century, stars variable in 
brilliancy have been known, and there are also stars which 
periodically change in color. The period of a variable star 
means the interval of time in which it goes through all its 
changes, and returns to the same brilliancy. 

The most noted variable stars are Mir a Ceti (o Ceti) 
and Algol (fi Persei). Mir a appears about twelve times 
in eleven years, and remains at its greatest brightness 
(sometimes as high as the 2d magnitude, sometimes not 
above the 4th) for some time, then gradually decreases for 
about 74 days, until it becomes invisible to the naked eye, 
and so remains for about five or six months. From the 
time of its reappearance as a lucid star till the time of its 
maximum is about 43 days (Heis). The mean period, or 
the interval from minimum to minimum, is about 333 
days (Arge lander), but this period, as does the maxi- 
mum light, varies greatly. 

Algol has been known as a variable star since 1667. Its 
period is about 2 d 20 h 49 ra , and is supposed to be from 
time to time subject to slight fluctuations. This star is 
commonly of the 2d magnitude ; after remaining so 
about 2J- days, it falls to 4 m in the short time of 4£ hours, 
and remains of 4 m for 20 minutes. It then commences 
to increase in brilliancy, and in another Z\ hours it is 



VARIABLE STARS. 



441 



again of the 2d magnitude, at which point it remains for 
the remainder of its period, about 2 d 12 h . 

These two examples of the class of variable stars give a 
rough idea of the extraordinary nature of the phenomena 
they present. A closer examination of others discloses 
minor variations of great complexity and apparently with- 
out law. 

The following are some of the more prominent vari- 
able stars visible to the naked eye : 



Name. 


R. A. 

1870. 


Declination, 
1870. 


Period. 


Changes of 
Magnitude. 


(3 Persei 

(5 Cephei 

7} Aquilae 

/? Lyrae 

a Herculis.. . . 

o Ceti 

v Hydrae 

ij Argus 


h. m. s. 

2 59 43 
22 24 21 
19 45 51 
18 45 17 
17 8 43 

2 12 47 
13 22 37 
10 40 2 


o / 

+ 40 27-2 
+ 57 45-0 
+ 40-4 
+ 33 12-7 
+ 14 32-4 

- 3 34-1 

- 22 36-4 
-59 0-1 


d. 

2-86 
5-36 
7-17 
12-91 
88-5 
330-0 
436-0 
70 years. 


from to 
2£ 4 
3-7 4-8 
3-6 4-7 

8+ h 

3-1 3-9 

2 10 
4 10 
1 6 



About 90 variable stars are well known, and as many 
more are suspected to vary. In nearly all cases the mean 
period can be fairly well determined, though anomalies of 
various kinds frequently appear. The principal anomalies 
are : 

First. The period is seldom constant. For some stars 
the changes of the period seem to follow a regular law ; 
for others no law can be fixed. 

Second. The time from a minimum to the next maxi- 
mum is usually shorter than from this maximum to the 
next minimum. 

Third. Some stars (as /> Lyrce) have not only one max- 
imum between two consecutive principal minima, but 
two such maxima. For fi Zyrce, according to Argelai*- 
der, 3 d 2 h after the principal minimum comes the first 
maximum ; then, 3 d 7 h after this, a secondary minimum in 
which the star is by no means so faint as in the principal 



442 



ASTRONOMY. 



minimum, and finally 3 d 3 h afterward conies the principal 
maximum, the whole period being 12 d 21 h 47 m . The 
course of one period is illustrated below, supposing the 
period to begin at d h , and opposite each phase is given 
the intensity of light in terms of y Lyrce = 1, according 
to photometric measures by Klein. 



Phase. 






Relative 
Intensity. 


Principal Minimum 

First Maximum 


0* 

3 d 


h 

2 h 

9 h 

12 h 

22 m 


0-40 
0-83 


Second Minimum 


6 d 


0-58 


Principal Maximum 


9 d 


0-89 


Principal Minimum 


12 d 


0-40 



The periods of 94 well-determined variable stars being 
tabulated, it appears that they are as follows : 



Period between 


No. of Stars. 


Period between 


No. of Stars. 


1 d. and 20 d. 
20 50 
50 100 
100 150 
150 200 
200 250 
250 300 
300 350 


13 
1 
4 
4 
5 
9 

14 
18 


350 d. and 400 d. 
400 450 
450 500 
500 550 
550 600 
600 650 
650 700 
700 750 


13 
8 
3 


1 

1 




2=94 



It is natural that there should be few known variables 
of periods of 500 days and over, but it is not a little re- 
markable that the periods of over half of these variables 
should fall between 250 and 450 days. 

The color of over 80 per cent of the variable stars is red 
or orange. Red stars (of which 600 to 700 are known) 
are now receiving close attention, as there is a strong like- 
lihood of finding among them many new variables. 

The spectra of variable stars show changes which ap- 
pear to be connected with the variations in their light. 



TEMPORAR Y STARS. 443 

Another class of variations occurs among the fixed stars— namely, 
variations in color, either with or without corresponding changes 
of magnitude. 

In the Uranometry, composed in the middle of the tenth century 
by the Persian astronomer Al Sufi, it is stated that at the time of 
his observations the star Algol was reddish — a term which he ap- 
plies also to the stars Antarcs, Aldebaran, and some others. Most 
of these still exhibit a reddish aspect. But Algol now appears as a 
white star, without any sign of color. Dr. Klein, of Cologne, 
discovered that a Ursce Majoris periodically changes color from an 
intense fiery red to a yellow or yellowish-red every five weeks. 
Weber, of Peckeloh, has observed this star lately, and finds this 
period to be well established. 

§ 2. TEMPORARY OR NEW STARS. 

There are a few cases known of apparently new stars 
which have suddenly appeared, attained more or less 
brightness, and slowly decreased in magnitude, either dis- 
appearing totally, or finally remaining as comparatively 
faint objects. 

The most famous one was that of 1572, which attained 
a brightness greater than that of Sirius or Jupiter and 
approached to Venus, being even visible to the eye in 
daylight. Tycho Bra he first observed this star in No- 
vember, 1572, and watched its gradual increase in light 
until its maximum in December. It then began to diminish 
in brightness, and in January, 1573, it was fainter than 
Jupiter. In February and March it was of the 1st mag- 
nitude, in April and May of the 2d, in July and August of 
the 3d, and in October and November of the 4th. It con- 
tinued to diminish until March, 1574, when it became in- 
visible, as the telescope was not then in use. Its color, 
at first intense white, decreased through yellow and red. 
"When it arrived at the 5th magnitude its color again 
became white, and so remained till its disappearance. 
Tycho measured, its distance carefully from nine stars near 
it, and near its place there is now a star of the 10th 
or 11th magnitude, which is possibly the same star. 

The history of temporary stars is in general similar to 
that of the star of 1572, except that none have attained so 



444 



ASTRONOMY. 



great a degree of brilliancy. More than a score of such 
objects are known to have appeared, many of them before 
the making of accurate observations, and the conclusion is 
probable that many have appeared without recognition. 
Among telescopic stars, there is but a small chance of de- 
tecting a new or temporary star. 

Several supposed cases of the disappearance of stars ex- 
ist, but here there are so many possible sources of error 
that great caution is necessary in admitting them. 

Two temporary stars have appeared since the invention 
of the spectroscope (1859), and the conclusions drawn 
from a study of their spectra are most important as throw- 
ing light upon the phenomena of variable stars in general. 

The first of these stars is that of 1866, called T Coronoe. 
It was first seen on the 12th of May, 1866, and was then 
of the 2d magnitude. Its changes were followed by vari- 
ous observers, and its magnitude found to diminish as 
follows : 



1866. 

[ay 12 


m. 

2-0 

2-2 


1866. 
May 18 


m. 
5-5 


13 


19 

20 


6-0 


14 


3-0 


6-5 


15 

16 


3-5 

4-0 

4-5 


21 

22 


7-0 

7-5 


17 


23 


8-0 



By June Tth it had fallen to 9 m, 0, and July 7th it was 
9 m -5. Schmidt's observations of this star (T Cor once), 
continued up to 1877, show that, after falling from the 
second to the seventh magnitude in nine days, its light 
diminished very gradually year after year down to nearly 
the tenth magnitude, at which it has remained pretty con- 
stant for some years. But during the whole period there 
have been fluctuations of brightness at tolerably regular 
intervals of ninety -four days, though of successively de- 
creasing extent. After the first sudden fall, there seems 
to have been an increase of brilliancy, which brought the 
star above the seventh magnitude again, in October, 
1866, an increase of a full magnitude ; but since that time 



VARIABLE STARS. 445 

the changes have been much smaller, and are now but 
little more than a tenth of a magnitude. The color of the 
star has been pale yellow throughout the whole course 
of observations. 

The spectroscopic observations of this star by Huggins and 
Miller showed it to have a spectrum then absolutely unique. The 
report of their observations says, " the spectrum of this object is 
twofold, showing that the light by which it shines has emanated 
from two distinct sources. The principal spectrum is analogous 
to that of the sun, and is formed of light which was emitted by 
an incandescent solid or liquid photosphere, and which has suffered 
a partial absorption by passing through an atmosphere of vapors at 
a lower temperature than the photosphere. Superposed over this 
spectrum is a second spectrum consisting of a few bright lines 
which is due to light which has emanated from intensely heated 
matter in the state of gas." 

In November, 1876, Dr. Schmidt discovered a new star in Cyg~ 
nus, whose telescopic history is similar to that given for T Coronm. 
When discovered it was of the 3d magnitude, and it fell rapidly 
below visibility to the naked eye. 

This new star in Cygnus was observed by Corntj, Copeland, and 
Vogel, by means of the spectroscope ; and from all the observa- 
tions it is plain that the hydrogen lines, at first prominent, have 
gradually i'aded. With the decrease in tneir brilliancy, a line 
corresponding in position with the brightest of the lines of a nebu- 
la has strengthened. On December 8th, 1876, this last line was much 
fainter than F (hydrogen line in the solar spectrum), while on 
March 2d, 1877, F was very much the fainter of the two. 

At first it exhibited a continuous spectrum with numerous bright 
lines, but in the latter part of 1877 it emitted only monochromatic 
light, the spectrum consisting of a single bright line, correspond- 
ing in position to the characteristic line of gaseous nebulas. The 
intermediate stages were characterized by a gradual fading out, 
not only of the continuous spectrum, but also of the bright lineG 
which crossed it. From this fact, it is inferred that this star, which 
has now fallen to 10 - 5 magnitude, has actually become a planetary 
nebula, affording an instance of a remarkable reversal of the pro- 
cess imagined by La Place in his nebular theory. 



§ 3. THEORIES OP VARIABLE STARS. 

The theory of variable stars now generally accepted by investi- 
gators is founded on the following general conclusions : 

(1) That the only distinction which can be made between the 
various classes of stars we have just described is one of degree. 
Between stars as regular as Algol, which goes through its period in 
less than three days, and the sudden blazing out of the star de* 



446 ASTRONOMY. 

scribed by Ttcho Brahe, there is every gradation of irregularity. 
The only distinction that can be drawn between them is in the 
length of the period and the extent and regularity of the changes. 
All such stars must, therefore, for the present, be included in the 
single class of variables. 

It was at one time supposed that newly created stars appeared 
from time to time, and that old ones sometimes disappeared from 
view. But it is now considered that there is no well-established 
case either of the disappearance of an old star or the creation of a 
new one. The supposed cases of disappearance arose from cata- 
loguers accidentally recording stars in positions where none existed. 
Subsequent astronomers finding no stars in the place concluded 
that the star had vanished when in reality it had never existed. 
The view that temporary stars are new creations is disproved by 
the rapidity with which they always fade away again. 

(2) That all stars may be to a greater or less extent variable ; 
only in a vast majority of cases the variations are so slight as to be 
imperceptible to the eye. If our sun could be viewed from the dis- 
tance of a star, or if we could actually measure the amount of light 
which it transmits to our eyes, there is little doubt that we should 
find it to vary with the presence or absence of spots on its surface. 
We are therefore led to the result that variability of light may be a 
common characteristic of stars, and if so we are to look for its 
cause in something common to all such objects. 

The spots on the sun may give us a hint of the probable cause of 
the variations iu the light of the stars. The general analogies of the 
universe, and the observations with the spectroscope, all lead us to 
the conclusion that the physical constitution of the sun and stars is 
of the same general nature. As we see spots on the sun which vary 
in form, size and number from day to day, so if we could take a suf- 
ficiently close view of the faces of the stars we should probably see 
spots on a great number of them. In our sun the spots never cover 
more than a very small fraction of the surface ; but we have no 
reason to suppose that this would be the case with the stars. If 
the spots covered a large j>ortion of the surface of the star, then 
their variations in number and extent would cause the star to vary 
in light. 

This view does not, however, account for those cases in which the 
light of a star is suddenly increased in amount hundreds of times. 
But the spectroscopic observations of T Goronm show another 
analogy with operations going on in our sun. Mr. Hugglns's ob- 
servations, which we have already cited, seem to show that there 
was a sudden and extraordinary outburst of glowing hydrogen 
from the star, which by its own light, as well as by heating up the 
whole surface of the star, caused an increase in its brilliancy. 

Now, we have on a very small scale something of this same kind 
going on in our sun. The red flames which are seen during a 
total eclipse are caused by eruptions of hydrogen from the interior 
of the sun, and these eruptions are generally connected with the 
faculse or portions of the sun's disk more brilliant than the rest of 
the photosphere. 



VARIABLE STARS. 447 

The general theory of variable stars which has now the most 
evidence in its favor is this : These bodies are, from some general 
cause not fully understood, subject to eruptions of glowing hydro- 
gen gas from their interior, and to the formation of dark spots on 
their surfaces. These eruptions and formations have in most cases 
a greater or less tendency to a regular period. 

In the case of our sun, the period is It years, but in the case of 
many of the stars it is much shorter. Ordinarily, as in the case of 
the sun and of a large majority of the stars, the variations are too 
slight to affect the total quantity of light to any visible extent. 
But in the case of the variable stars this spot-producing power and 
the liability to eruptions are very much greater than in the case of 
our sun, and thus we have changes of light which can be readily 
perceived by the eye. Some additional strength is given to this 
theory by the fact just mentioned, that so large a proportion of 
the variable stars are red. It is well known that glowing bodies 
emit a larger proportion of red rays and a smaller proportion of 
blue ones the cooler they become. It is therefore probable that 
the red stars have the least heat. This being the case, it is more 
easy to produce spots on their surface ; and if their outside surface 
is so cool as to become solid, the glowing hydrogen from the in- 
terior when it did burst through would do so with more power 
than if the surrounding shell were liquid or gaseous. 

There is, however, one star of which the variations may be due to 
an entirely different cause — namely, Algol. The extreme regularity 
with which the light of this object fades away and disappears sug- 
gests the possibility that a dark body may be revolving around it, 
and partially eclipsing it at every revolution. The law of variation 
of its light is so different from that of the light of other variable 
stars as to suggest a different cause. Most others are near their 
maximum for only a small part of their period, while Algol is at its 
maximum for nine tenths of it. Others are subject to nearly con- 
tinuous changes, while the light of Algol remains constant during 
nine tenths of its period. 



CHAPTER III. 

MULTIPLE STARS. 

§ 1. CHARACTER OF DOUBLE AND MULTIPLE 

STARS. 

When we examine the heavens with telescopes, we find 
many cases in which two or more stars are extremely close 
together, so as to form a pair, a triplet, or a group. It is 
evident that there are two ways to account for this ap- 
pearance. 

1. We may suppose that the stars happen to lie nearly 
in the same straight line from us, but have no connection 
with each other. It is evident that in this case a pair of 
stars might appear double, although the one was hundreds 
or thousands of times farther off than the other. It is, 
moreover, impossible, from mere inspection, to determine 
which is the farther off. 

2. We may suppose that the stars are really as near 
together as they appear, and are to be considered as form- 
ing a connected pair or group. 

Couples of stars in the first case are said to be optically 
double, and are not generally classed by astronomers as 
double stars. 

Stars which are considered as really double are those 
which are so near together that we are justified in consider- 
ing them as physically connected. Such stars are said to 
be physically double, and are generally designated as 
double stars simply. 

Though it is impossible by mere inspection to decide to* 
which class a pair of stars should be considered as belong 
ing, yet the calculus of probabilities will enable us to de 



DOUBLE STARS. 



449 



cide in a rough way whether it is likely that two stars not 
physically connected should appear so very close together 
as most of the double stars do. This question was first 
considered by the Kev. John Michell, F.B.S., of Eng- 
land, who in 1777 published a paper on the subject in the 
Philosophical Transactions. He showed that if the lucid 
stars were equally distributed over the celestial sphere, the 
chances were 80 to 1 against any two being within three 
minutes of each other, and that the chances were 500,000 
to 1 against the six visible stars of the Pleiades being 
accidentally associated as we see them. When the mill- 
ions of telescopic stars are considered, there is a greater 
probability of such accidental juxtaposition. But the 
probability of many such cases occurring is so extremely 
small that astronomers regard all the closest pairs as phy- 
sically connected. It is now known that of the 600,000 
stars of the first ten magnitudes, at least 10,000, or one out 
of every 60, has a companion within a distance of 30" of 
arc. This proportion is many times greater than could 
possibly be the result of chance. 

There are several cases of stars which appear double to 
the naked eye. Two of these we have already described 
— namely, 6 Tauri and £ Lyrce. The latter is a most 
curious and interesting object, from the fact that each of 
the two stars which compose it is 
itself double. No more striking 
idea of the power of the teles- 
cope can be formed than by 
pointing a powerful instrument 
upon this object. It will then 
be seen that this minute pair of 
points, capable of being distin- 
guished only by the most perfect 

eye, is really composed of two Fig. 122.— the quadruple 
pairs of stars wide apart, with a STAIi £ L ™- 

group of smaller stars between and around them. The 
figure shows the appearance in a telescope of considerable 
power. 




450 



ASTRONOMY. 



Revolutions of Double Stars— Binary Systems. — The 
most interesting question suggested by double stars is that 
of their relative motion. It is evident that if these 
bodies are endowed with the property of mutual gravita- 
tion, they must be revolving around each other, as the 
earth and planets revolve around the sun, else they would 
be drawn together as a single star. With a view of detect- 
ing this revolution, astronomers measure the position- 
angle, and distance of these objects. The distance of the 




Fig. 123. — measurement op position-angle. 

components of the double star is simply the apparent 
angle which separates them, as seen by the observer. It is 
always expressed in seconds or fractions of a second of arc. 
The angle of position, or " position-angle" as it is often 
called for brevity, is the angle which the line joining the 
two stars makes with the line drawn from the brightest star 
to the north pole. If the fainter star is directly north of 
the brighter one, this angle is zero ; if east, it is 90°; if south, 



DOUBLE STARS. 



451 



The two latter words refer 



it is 180° ; if west, it is 270°. This is illustrated by the 
figure, which is supposed to represent the Held of view of 
an inverting telescope pointed toward the south. The 
arrow shows the direction of the apparent diurnal motion. 
The telescope is supposed to be so pointed that the brighter 
star may be in the centre of the field. The numbers 
around the surrounding circle then show the angle of po- 
sition, supposing the smaller star to be in the direction of 
the number. 

The letters sp, sf, np, and nf show the methods of 
dividing the four quadrants, s meaning south, n north. 
f following, and^? preceding, 
to the direction of the diur- 
nal motion. Fig. 124 is an 
example of a pair of stars in 
which the position-angle is 
about 44°. 

If, by measures of this 
sort extending through a 
series of' years, the distance 
or position-angle of a pair 
of stars is found to change, 
it shows that one star is re- 
volving around the other. 
Such a pair is called a 
binary star or "binary sys- 
tem. The onlv distinction 
which we can make between 
binary systems and ordinary double stars is founded on 
the presence or absence of observed motion. It is prob- 
able that nearly all the double stars are really binary sys- 
tems, but that many thousands of years are required to 
perform a revolution, so that the motion has not yet been 
detected. 

The discovery of binary systems is one of great scien- 
tific interest, because from them we learn that the law ot 
gravitation includes the stars as well as the solar system in 




Fig. 



124. — POSITION- ANGLE OP A 
DOUBLE STAR. 



452 ASTRONOMY. 

its scope, and may therefore be regarded as a universal 
property of matter. 

Colors of Double Stars. — There are a few noteworthy statistics 
in regard to the colors of the components of double stars which 
may be given. Among 596 of the brighter double stars, there are 
375 pairs where each component has the same color and intensity ; 
101 pairs where the components have same color, but different in- 
tensity ; 120 pairs of different colors. Among those of the same 
color, the vast majority were both white. Of the 476 stars of the 
same color, there were 295 pairs whose components were both 
white ; 118 pairs whose components were both yellow or both red ; 
63 pairs whose components were both bluish. When the com- 
ponents are of different colors, the brighter generally appears to 
have a tinge of red or yellow ; the other of blue or green. 

These data indicate in part real physical laws. They also are 
partly due to the physiological fact that the fainter a star is, the 
more blue it will appear to the eye. 

Measures of Double Stars. — The first systematic measures of 
the relative positions of the components of double stars were made 
by Christian Mayer, Director of the Ducal Observatory of Mann- 
heim, 1778, but it is to Sir William Herschel that we owe the ba- 
sis of our knowledge of this branch of sidereal astronomy. In 1780 
Herschel measured the relative situation of more than 400 double 
stars, and after repeating his measures some score of years later, 
he found in about 50 of the pairs evidence of relative motion of 
the components. In this first survey he found 97 stars whose dis- 
tance was under 4", 102 between 4" and 8", 114 between 8" and 
16", and 132 between 16" and 32". 

Since Herschel's observations, the discoveries of Sir John Her- 
schel, Sir James South, Dawes, and many others in England, of 
W. Struve, Otto Struve, Madler, Secchi, Dembowski, Du- 
ner, in Europe, and of G. P. Bond, Alvan Clark, and S. W. 
Burnham, in the United States, have increased the number of 
known double stars to about 10,000. 

Besides the double stars, there are also triple, quadruple, etc., 
stars. These are generically called multiple stars. The most re- 
markable multiple star is the Trapezium, in the centre of the nebula 
of Orion, commonly called Orionis, whose four stars are, without 
doubt, physically connected. 

The next combination beyond a multiple star is a cluster of stars ; 
and beginning with clusters of 1' in diameter, such objects may be 
found up to 30' or more in diameter, every intermediate size being 
represented. These we shall consider shortly. 

§ 2. ORBITS OF BINARY STARS. 

When it was established that many of the double stars were really 
revolving around each other, it became of great interest to 
determine the orbit and ascertain whether it was an ellipse, with 



BINARY STARS. 



453 



the centre of gravity of the two objects in one of the foci ; if so, it 
would be shown that gravitation among the stars followed the same 
law as in the solar system. As an illustration of how this may be 
done, we present the following measures of the position-angle and 
distance of the binary star £ Ursm Majoris, which was the first one 
of which the orbit was investigated. The following notation is 
used : p, the angle of position ; s, the distance ; A, the brighter 
star ; B, the fainter one. 

£ UrszE Majoris = 2 1523.* 



Epoch. 


P 


s. 


Observer. 


1782-0 


© 

143-8 

97-5 

276-4 

264-7 
201-1 
150-9 
122-6 
96-7 
16-5 


1*92 
1-90 
2-45 
2-99 
2-56 
0-91 


W. Herschel. 


1802 • 1 




1820-1 

1821-8 


W. Struve. 
<« 


1831-3 


J. Herschel. 


1840-3 


Dawes. 


1851-6. 

1863-2 


Madler. 
Dembowski. 


1872-5 


Duner. 







If these measures be plotted on a sheet of squared paper, the 
several positions of B will be found to lie in an ellipse. This ellipse 
is the projection of the real orbit on the plane perpendicular to the 
line of sight, or line joining the earth with the star A. It is a 
question of analysis to determine the true orbit from the times and 
from the values of p and s. 

If the real orbit happened to lie in a plane perpendicular to the 
line of sight, the star A would lie in the focus of the ellipse. If 
this coincidence does not take place, then the plane of the true or- 
bit is seen obliquely. 

The first two of Kepler's laws can be employed in determining 
such orbits, but the third law is inapplicable. 

Masses of Binary Systems. — When the parallax or distance, 
the semi-major axis of the orbit, and the time of revolution of a 
binary system are known, we can determine the combined mass of 
the pair of stars in terms of the mass of the sun. Let us put : 

a", the mean distance of the two components as measured in 
seconds ; 

a, their mean distance from each other in astronomical units ; 

T, the time of revolution in years ; 

Mj Mo, the masses of the two component stars ; 

P, their annual parallax ; 

D, their distance in astronomical units. 



* 2 1523 signifies that this star is No. 1523 of W. Struve's Dorsal 
Catalogue. 



454 ASTRONOMY. 

From the generalization of Kepler's third law, given by the 
theory of gravitation, we have 

From the formulae explained in treating of parallax we have 

D = 1 -f- sin. P. 

If a" is the major axis in seconds, a being the same quantity in 
astronomical units, then 

a = D - sin. a\ 

From these two equations, 

_ sin. a" _ a" 
a ~ ^n7p-~P~* 

because a" and P are so small that the arcs may be taken for their 
sines. 

Putting this value of a in the equation for M •+■ M , 

we have M + M = 



T 2 P 3 



a Centauri and p Ophiuclii are two binary stars whose parallaxes 
have heen determined (0"-98 and 0"-16) from direct measures. For 
a. Centauri 

T - 77-0 years; a" = 15 "-5 ; P = 0"-98; 

forp OpJtiuchi, 

T - 94-4 years; a" = 4" -70; P — 0"-16. 

If we substitute in the last equation these values for T, P, and a" t 
we have 

Mo + M — • 67 for a Centauri, 
Mo 4- M = 2 • 84 f or p Ophiuchi. 

The last number is quite uncertain, owing to the difficulty of meas- 
uring so small a parallax. We can only conclude that the mass of 
these two systems is not many times greater or less than the mass of 
our sun. From the agreement in these two cases, it is probable that 
in other systems, if the mass could be determined, it would not be 
greatly different from the mass of our sun We may on this supposi- 
tion, which amounts to supposing J/ + M= 1, apply the formula 

to other binaries, and deducn a value for Pin each case which is called 
the hypothetical parallax (Gylden), and which is probably not far 
from the truth. 

There are, beside binary systems, multiple ones as f Cancri, where 
the distance of A and B is 0"-8 ; and from the middle point between 

A 4- B 
A and B to Cis 5" -5. The period of revolution of — - — about C is 

Z 
supposed to be about 730 years. If in the last formula we put 
T = 730 years and a" = 5" -5, we have the hypothetical parallax 

p = 0"-062. 



BINARY STABS. 



455 



Following are given the elements of several of the more impor- 
tant binary stars. Eight of these have moved through an entire 
revolution — 360° — since the first observation, and about 150 are 
known which have certainly moved through an arc of over 10° since 
they were first observed. 

In the tables the semi-major axis, or mean distance, must be 
given in seconds, since we have usually no data by which its value 
in linear measures of any kind can be fixed. 

Periods of revolution exceeding 120 years must be regarded as 
quite uncertain* 

Elements op Binary wtars. 



Star's Name. 



Period 
(Years.) 



42 Comae Ber. . . 

£ Herculis 

2 3121* 

j] Coronae Bor. . 

k Librae 

y Coronae Aus. . 

£ Ursae Maj. 

£ Cancri 

a Centauri 

70 Ophiuchi. . . . 
y Coronae Bor. . . 

S062 2.. 

u Leonis 

"k Ophiuchi 

p Eridani 

1768 2 

4 Bootis 

7 Virginis 

t Ophiuchi 

ri Cassiopeae 

44 Bootis 

1938 2 ) 

(U 2 Bootis ) 

36 Andromeda.. 

7 Leonis 

6 Cygni 

61 Cygni 

c Coronae Bor 

a Geminorum. . , 
C Aquarii 



25 

34 

37 

40 

95 

55 

60 

60 

62 

60 

85 

92 

95 

104 

114 

233 

117 

124 

127 

175 

217 

222 

261 

280 

349 
402 
415 
452 
845 
1001 
1578 



Time 
of Peri- 


Semi- 
Axis 


Eccen- 


Calculator. 


astron. 


Major. 






1869-9 


0"-65 


0-48 


Dubiago. 


1864-9 


1-36 


0-41 


Flammarion. 


1842-8 


[0-71] 


0-26 


Doberck. 


1849-9 


0-99 


0-29 


Flammarion. 


1859-6 


1-26 


0-08 


Doberck. 


1882-7 


2-40 


0-69 


Schiaparelli. 


1875.6 


2-58 


0-38 


Hind. 


1875-5 


2-54 


0-37 


Flammarion. 


1869-3 


0-90 


0-00 


O. Struve. 


1869.9 


0-91 


0-37 


Flammarion. 


1874-9 


21-80 


0-67 


Hind. 


1807-9 


4-88 


0-39 


Flammarion. 


1843-7 


0-70 


0-35 


Doberck. 


1834-9 


1-27 


0-46 


Doberck. 


1841-6 


0-85 


0-55 


Doberck. 


1803-9 


1-19 


0-49 


Doberck. 


1817-5 


3-82 


0-38 


Doberck. 


1863-0 




0-66 


Doberck. 


1770-7 


4-86 


0-71 


Doberck. 


1836-5 


3-39 


0-87 


Flammarion. 


1821-9 


1-40 


0-61 


Doberck. 


1909-2 


9-83 


0-57 


Doberck. 


1783-0 


3-09 


0-71 


Doberck. 


1863-5 


1-47 


0-60 


Doberck. 


1798-8 


1-54 


0-65 


Doberck. 


1741-1 


2-00 


0-74 


Doberck. 


1904-1 


2-31 


0-2S 


Behrmann. 




15-4 
5-89 


6 '-75 




1826-9 


Doberck. 


1749-8 


7-43 


0-33 


Doberck. 


1924-2 


7-64 


0-65 


Doberck. 



3121 2 signifies No. 3121 of W- Struve's Dorpat Catalogue. 



456 ASTRONOMY. 

The first computation of the orbit of a binary star was made by 
Savary (Astronomer at the Paris Observatory) about 1826, and his 
results were the first which demonstrated that the laws of gravita 
tion, which we knew to be operative over the extent of the solar 
system, and even over the vast space covered by the orbit of 
Halley's comet, extended even further, to the fixed stars. It might 
have been before 1825 a hazardous extension of our views to sup- 
pose even the nearest fixed stars to be subject to the laws of New- 
ton ; but as many of the known binaries have no measurable paral- 
lax, it is by no means an unsafe conclusion that every fixed star 
which our best telescopes will show is subjected to the same laws 
as those which govern the fall of bodies upon the earth. 



CHAPTER IV. 

NEBULAE AND CLUSTERS. 
§ 1. DISCOVERY OF NEBULiE. 

In the star-catalogues of Ptolemy, Hevelius and the 
earlier writers, there was included a class of nebulous or 
cloudy stars, which were in reality star-clusters. They 
appeared to the naked eye as masses of soft diffused light 
of greater or less extent. In this respect, they were quite 
analogous to the Milky Way. When Galileo first di- 
rected his telescope upon them their nebulous appearance 
vanished, and they were seen to consist of clusters of 
stars. 

As the telescope was improved, great numbers of such 
patches of light were found, some of which could be re- 
solved into stars, while others could not. The latter were 
called nebulce and the former star- clusters. 

About 1656, Huyghens described the great nebula of 
Orion, one of the most remarkable and brilliant of these 
objects. During the last century, Messier, of Paris, made 
a list of 103 northern nebulge, and Lacaille noted a few of 
those of the southern sky. The careful sweeps of the 
heavens by Sir William Hekschel with his great tele- 
scopes first gave proof of the enormous number of these 
masses. In 17S6, he published a catalogue of one thousand 
new nebulae and clusters. This was followed in 17S9 by 
a catalogue of a second thousand, and in 1802 by a third 
catalogue of five hundred new objects of this elass. A 



458 ASTRONOMY. 

similar series of sweeps, carried on by Sir John Her- 
schel in both hemispheres, added about two thousand 
more nebulae. The general catalogue of nebulae and clus- 
ters of stars of the latter astronomer, published in 1864, 
contains 5079 nebulae : 6251 are known in 1879. Over 
two thirds of these were iirst discovered by the Herschels. 
The mere enumeration of over 4000 nebulae is, how- 
ever, but a small part of the labor done by these two dis- 
tinguished astronomers. The son has left a great number 
of studies, drawings, and measures of nebulae, and the 
memoirs of the father on the Construction of the Heavens 
owe their suggestiveness and much of their value to his 
long-continued observations on this class of objects, which 
gave him the clue to his theories. 

§ 2. CLASSIFICATION OF NEBULA AND CLUSTERS. 

In studying these objects, the iirst question we meet is 
this : Are all these bodies clusters of stars which look 
diffused only because they are so distant that our tele- 
scopes cannot distinguish them separately ? or are some of 
them in reality what they seem to be — namely, diffused 
masses of matter ? 

In his early memoirs of 1784 and 1785, Sir William: 
Herschel took the first view. He considered the Milky 
Way as nothing but a congeries of stars, and all nebulae- 
naturally seemed to him to be but stellar clusters, so 
distant as to cause the individual stars to disappear in a 
general milkiness or nebulosity. 

In 1791, however, his views underwent a change. He 
had discovered a nebulous star (properly so called), or a 
star which was undoubtedly similar to the surrounding' 
stars, and which was encompassed by a halo of nebulous- 
light. * 

* This was the 69th nebula of his fourth class of planetary nebul»~ 
(H. iv. 69.) 



NEBULA AND CLUSTERS. 459 

He says : " Nebulas can be selected so that an insensible grada- 
tion shall take place from a coarse cluster like the Pleiades down to 
a milky nebulosity like that in Orion, every intermediate step being 
represented. This tends to confirm the hypothesis that all are com- 
posed of stars more or less remote. 

lt A comparison of the two extremes of the series, as a coarse 
cluster and a nebulous star, indicates, however, that the nebulosity 
about the star is not of a starry nature. 

" Considering H, iv. 69, as a typical nebulous star, and supposing 
the nucleus and chevelure to be connected, we may, first, sujDpose 
the whole to be of stars, in which case either the nucleus is enor- 
mously larger than other stars of its stellar magnitude, or the envelope 
is composed of stars indefinitely small ; or, second, we must admit 
that the star is involved in a shining fluid of a nature totally unknown 
to us. 

"The shining fluid might exist independently of stars. The 
light of this fluid is no kind of reflection from the star in the cen- 
tre. If this matter is self-luminous, it seems more fit to produce a 
star by its condensation than to depend on the star for its existence. 

" Both diffused nebulosities and planetary nebulae are better 
accounted for by the hypothesis of a shining fluid than by suppos- 
ing them to be distant stars." 

This was the first exact statement of the idea that, beside 
stars and star-clusters, we have in the universe a totally 
distinct series of objects, probably much more simple in 
their constitution. The observations of Httggins and 
Secchi on the spectra of these bodies have, as we shall 
see, entirely confirmed the conclusions of Hekschel. 

Nebulae and clusters were divided by Hekschel into 
classes. Of his names, only a few are now in general use. 
He applied the name planetary nebulae to certain circular 
or elliptic nebulae which in his telescope presented disks 
like the planets. Spiral nebulce are those whose convo- 
lutions have a spiral shape. This class is quite numer- 
ous. 

The different kinds of nebulae and clusters will be better under- 
stood from the cuts and descriptions which follow than by formal 
definitions. It must be remembered that there is an almost infinite 
variety of such shapes. 

The figure by Sir John Hekschel on the next page gives a good 
idea of a spiral or ring nebula. It has a central nucleus and a small 
and bright companion nebula near it. In a larger telescope than 
IlERscnEL's its aspect is even more complicated. "See also Fig. 128. 



460 



ASTRONOMY. 



The Omega or horseshoe nebula, so called from the resemblance 
of the brightest end of it to a Greek £2, or to a horse's iron shoe, is 
one of the most complex and remarkable of the nebulas visible in 
the northern hemisphere. It is particularly worthy of note, as 
there is some reason to believe that it has a proper motion. Cer- 
tain it is that the bright star which in the figure is at the left-hand 
upper corner of one of the squares, and on the left-hand (west) 
edge of the streak of nebulosity, was in the older drawings placed 
on the other side of this streak, or within the dark bay, thus mak- 
ing it at least probable that either the star or the nebula has moved. 




Fig. 125.— spiral nebula. 

The trijid nebula, so called on account of its three branches 
which meet near a central dark space, is a striking object, and 
was suspected by Sir John Herschel to have a proper motion. 
Later observations seem to confirm this, and in particular the three 
bright stars on the left-hand edge of the right-hand (east) mass are 
now more deeply immersed in the nebula than they were observed 
to be by Herschel (1833) and Mason, of Yale College (1837). In 
1784, Sir William Herschel described them as " in the middle of 
the [dark] triangle." This description does not apply to their 
present situation. (Fig. 127j. 







FtO. 120. — THE OMEGA OR HORSESIIOE NEBULA. 



462 



ASTRONOMY. 



§ 3. STAR CLUSTERS. 

The most noted of all the clusters is the Pleiades, which have 
already been briefly described in connection with the constellation 
Taurus. The average naked eye can easily distinguish six stars 
within it, but under favorable conditions ten, eleven, twelve, or 




Fig. 127. — the trifid nebula. 



With the telescope, over a hundred 



A view of these is given in the map accompanying 



more stars can be counted, 
stars are seen 

jhe description of the Pleiades, Fig. 113, p. 425. This group con- 
tains Tempel's variable nebula, so called because it has been sup- 
posed to be subject to variations of light. This is probably not a 
variable nebula. 



NEBULJE AND CLUSTEBb. 



463 



The clusters represented in Figs. 129 and 130 are good examples 
of their classes. The first is globular and contains several thousand 
small stars. The central regions are densely packed with stars, 
and from these radiate curved hairy-looking branches of a spiral 
form. The second is a cluster of about 200 stars, of magnitudes 
varying from the ninth to the thirteenth and fourteenth, in which 
the brighter stars are scattered in a somewhat unusual manner 




FlO. 128. — THE RTNG NEBULA IN LYRA. 

over the telescopic field. This cluster is an excellent example of 
the " compressed " form so frequently exhibited. In clusters of 
this class the spectroscope shows that each of the individual stars 
is a true sun, shining by its native brightness. If we admit that a 
cluster is real — that is, that we have to do with a collection of stars 
physically connected — the globular clusters become important. It 
is a fact of observation that in general the stars composing such 



464: 



ASTRONOMY. 



clusters are about of equal magnitude, and are more condensed at 
the centre than at the edges. They are probably subject to central 
powers or forces. This was seen by Sir William Herschel in 1789. 
He says : 

" Not only were round nebulae and clusters formed by central 
powers, but likewise every cluster of stars or nebula that shows a 
gradual condensation or increasing brightness toward a centre. 
This theory of central power is fully established on grounds of ob- 
servation which cannot be overturned. 

" Clusters can be found of 10' diameter with a certain degree of 
compression and stars of a certain magnitude, and smaller clusters 
of 4', 3' or 2' in diameter, with smaller stars and greater compression, 
and so on through resolvable nebulae by imperceptible steps, to the 
smallest and faintest [and most distant] nebulae. Other clusters 




Fig. 129.— globular cluster. Fig. 130.— compressed cluster. 



there are, which lead to the belief that either they are more com- 
pressed or are composed of larger stars. Spherical clusters are 
probably not more different in size among themselves than different 
individuals of plants of the same species. As it has been shown 
that the spherical figure of a cluster of stars is owing to central 
powers, it follows that those clusters which, cceteris paribus, are the 
most complete in this figure must have been the longest exposed 
to the action of these causes. 

" The maturity of a sidereal system may thus be judged from 
the disposition of the component parts. 

" Though we cannot see any individual nebula pass through all 
its stages of life, we can select particular ones in each peculiar 
stage, ' ' and thus obtain a single view of their entire course of de- 
velopment. 



NEBULJS. 465 



% 4. SPECTRA OF NEBULA AND CLUSTERS. 

In 1864, five years after the invention of the spectroscope, Dr. 
Huggins, of London, commenced the examination of the spectra 
of the nebulae, and was led to the discovery that while the spectra 
of stars were invariably continuous and crossed with dark lines 
similar to those of the solar spectrum, those of many nebulae were 
discontinuous, showing these bodies to be composed of glowing gas. 
The figure shows the spectrum of one of the most famous planetary 
nebulae. (H. iv. 37.) The gaseous nebulae include nearly all the 
planetary nebulae, and very frequently have stellar-like condensa- 
tions in the centre. 

Singularly enough, the most milky looking of any of the nebulae 
(that in Andromeda) gives a continuous spectrum, while the nebula 
of Orion, which fairly glistens with small stars, has a discontinuous 





& 6 





Fig. 131. — spectrum of a planetary nebula. 

spectrum, showing it to be a true gas. Most of these stars are too 
faint to be separately examined with the spectroscope, so that we 
cannot say whether they have the same spectrum as the nebulae. 

The spectrum of most clusters is continuous, indicating that the 
individual stars are truly stellar in their nature. In a few cases, 
however, clusters are composed of a mixture of nebulosity (usually 
near their centre) and of stars, and the spectrum in such cases is 
compound in its nature, so as to indicate radiation both by gaseous 
and solid matter. 



§ 5. DISTRIBUTION OP NEBULIZE AND CLUSTERS 
ON THE SURFACE OF THE CELES- 
TIAL SPHERE. 

The following map (Fig. 132) by Mr. R. A. Proctor, gives at a 
glance the distribution of the nebulae on the celestial sphere with 
reference to the Milky Way, whose boundaries only are indicated 



STAR. CL USTEBS. 467 

The position of each nebula is marked by a dot ; where the dots are 
thickest there is a region rich in nebulae. A casual examination 
shows that such rich regions are distant from the Galaxy, and it 
would appear that it is a general law that the nebulse are distri- 
buted in greatest number around the two poles of the galactic 
circle, and that in a general way their number at any point of the 
sphere increases with their distance from this circle. This was 
noticed by the elder Herschel, who constructed a map similar to 
the one given. It is precisely the reverse of the law of apparent 
•distribution of the true star-clusters, which in general lie in or near 
the Milky Way. 



CHAPTER V. 

SPECTRA OF FIXED STARS. 

1. CHARACTERS OP STELLAR SPECTRA. 

Soon after the discovery of the spectroscope, Dr. Huggins and 
Professor W. A. Miller applied this instrument to the examina- 
tion of stellar spectra, which were found to be, in the main, similar 
to the solar spectrum — i.e., composed of a continuous band of the 
prismatic colors, across which dark lines or bands were laid, the 
latter being fixed in position. These results showed the fixed stars 
to resemble our own sun in general constitution, and to be com- 
posed of an incandescent nucleus surrounded by a gaseous and 
absorptive atmosphere of lower temperature. This atmosphere 
around many stars is different in constitution from that of the sun, 
as is shown by the different position and intensity of the various- 
black lines and bands. 

The various stellar spectra have been classified by Secchi into 
four types, distinguished from one another by marked differences in 
the position, character, and number of the dark lines. 

Type I is composed of the white stars, of which Sirius and Vega 
are examples (the upper spectrum in the plate Fig. 133). The spec- 
trum of these stars is continuous, and is crossed by four dark 
lines, due to the presence of large quantities of hydrogen in 
the envelope. Sodium and magnesium lines are also seen, and 
others yet fainter. 

Type II is composed mainly of the yellow stars, like our own 
sun, Arcturus, Capella, Aldebaran, and Pollux. The spectrum of 
the sun is shown in the second place in the plate. The vast ma- 
jority of the stars visible to the naked eye belong to this class. 

Type III (see the third and fourth spectra in the plate) is com- 
posed of the brighter reddish stars like a Orionis, Antares, a Herculis, 
etc. These spectra are much contracted toward the violet end, and 
are crossed by eight or more dark bands, these bands being them- 
selves resolvable into separate lines. 

These three types comprise nearly all the lucid stars, and it is 
not a little remarkable that the essential differences between the 
three classes were recognized by Sir William Herschel as early 
as 1798, and published in 1814. Of course his observations were 
made without a slit to his spectroscopic apparatus. 



blELLAR SPECTRA. 



469 




470 ASTRONOMY. 

Type IV comprises the red stars, which are mostly telescopic. 
The characteristic spectrum is shown in the last figure of the plate. 
It is curiously banded with three bright spaces separated by 
darker ones. 

It is probable that the hotter a star is the more simple a spectrum 
it has ; for the brightest, and therefore probably the hottest stars, 
such as Sirius, give spectra showing only very thick hydrogen lines 
and a few very thin metallic lines, while the cooler stars, such as 
our sun, are shown by their spectra to contain a much larger num- 
ber of metallic elements than stars of the type of Sirius, but no 
non-metallic elements (oxygen possibly excepted). The coolest 
stars give band-spectra characteristic of compounds of metallic 
with non-metallic elements, and of the non-metallic elements un- 
combined. 



§ 2. MOTION OF STARS IN THE LINE OP SIGHT. 

Spectroscopic observations of stars not only give information in 
regard to their chemical and physical constitution, but have been 
applied so as to determine approximately the velocity in kilometres 
per second with which the stars are approaching to or receding- 
from the earth along the line joining earth and star. The theory 
of such a determination is briefly as follows : 

In the solar spectrum we find a group of dark lines, as a, b, c, 
which always maintain their relative position. From laboratory 
experiments, we can show that the three bright lines of incandescent 
hydrogen (for example) have always the same relative position as 
the solar dark lines a, b, c. From this it is inferred that the solar 
dark lines are due to the presence of hydrogen in its absorptive 
atmosphere. 

Now, suppose that in a stellar spectrum we find three dark 
lines a', b', c', whose relative position is exactly the same as that 
of the solar lines a, b, c. Not only is their relative position the 
same, but the characters of the lines themselves, so far as the fainter 
spectrum of the star will allow us to determine them, are also simi- 
lar — that is, a' and a, b' and b, c' and c are alike as to thickness, 
blackness, nebulosity of edges, etc., etc. From this it is inferred 
that the star really contains in its atmosphere the substance whose 
existence has been shown in the sun. 

If we contrive an apparatus by which the stellar spectrum is seen 
in the lower half (say) of the eye-piece of the spectroscope, while 
the spectrum of hydrogen is seen just above it, we find in some 
cases this remarkable phenomenon. The three dark stellar lines, 
a', b', c', instead of being exactly coincident with the three hydro- 
gen lines a, b, c, are seen to be all thrown to one side or the 
other by a like amount — that is, the whole group a, b', c\ while 
preserving its relative distances the same as those of the compari- 
son group a, Z>, c, is shifted toward either the violet or red end of 
the spectrum by a small yet measurable amount. Repeated experi- 



STELLAR SPECTRA 



471 



ments by different instruments and observers show always a shifting 
in the same direction and of like amount. The figure shows the 
shifting of the F line in the spectrum of Sirius, compared with one 
fixed line of hydrogen. 

This displacement of the 
spectral lines is now ac- 
counted for by a motion of 
the star toward or from the 
earth. It is shown in Phy- 
sics that if the source of 
the light which gives the 
spectrum a', h\ c is mov- 
ing away from the earth, this 
group will be shifted toward 
the red end of the spec- 
trum ; if toward the earth, 
then the whole group will 
be shifted toward the blue 
end. The amount of this 
shifting is a function of the 
velocity of recession or ap- 
proach, and this velocity in 
miles per second can be 
calculated from the meas- 
ured displacement. This has been done for many stars by Dr. 
Huggins, Dr. Vogel, and Mr. Christie. Their results agree well, 
when the difficult nature of the research is considered. The rates 
of motion vary from insensible amounts to 100 kilometres per sec- 
ond ; and in some cases agree remarkably with the velocities com- 
puted from the proper motions and probable parallaxes. 




Fig. 134. 



-F-LINE IN SPECTRUM OP 
SIRIUS. 



CHAPTER VI. 

MOTIONS AND DISTANCES OF THE STAES. 

I 

§ 1. PROPER MOTIONS. 

We have already stated that, to the unaided vision,, the 
fixed stars appear to preserve the same relative position in 
the heavens through many centuries, so that if the an- 
cient astronomers could again see them, they could hardly 
detect the slightest change in their arrangement. But 
the refined methods of modern astronomy, in which the 
power of the telescope is applied to celestial measurement, 
have shown that there are slow changes in the positions 
of the brighter stars, consisting in a motion forward in a 
straight line and with uniform velocity. These motions 
are, for the most part, so slow that it would require thou- 
sands of years for the change of position to be percepti- 
ble to the unaided eye. They are called proper motions. 

As a general rule, the fainter the stars the smaller the proper mo- 
tions. For the most part, the proper motions of the telescopic stars 
are so minute that they have not been detected except in a very 
few cases. This arises partly from the actual slowness of the mo- 
tion, and partly from the fact that the positions of these stars have 
not generally been well determined. It will be readily seen that, in 
order to detect the proper motion of a star, its position must be de- 
termined at periods separated by considerable intervals of time. 
Since the exact determinations of star positions have only been 
made since the year 1750, it follows that no proper motion can be 
detected unless it is large enough to become perceptible at the end 
of a century and a quarter. With very few exceptions, no accurate 
determination of the positions of telescopic stars was made until 
about the beginning of the present century. Consequently, we 
cannot yet pronounce upon the proper motions of these stars, and 



MOTIONS OF THE STAMS. 473 

can only say that, in general, they are too small to be detected by 
the observations hitherto made. 

To this rule, that the smaller stars have no sensible proper mo- 
tions, there are a few very notable exceptions. The star Groom- 
bridge 1830, is remarkable for having the greatest proper motion of 
any in the heavens, amounting to about 7" in a year. It is only of 
the seventh magnitude. Next in the order of proper motion comes 
the double star 61 Cygni, which is about of the fifth magnitude- 
There are in all seven small stars, all of which have a larger proper 
motion than any of the first magnitude. But leaving out these ex- 
ceptional cases, the remaining stars show, on an average, a diminu- 
tion of proper motion with brightness. In general, the proper 
motions even of the brightest stars are only a fraction of a second 
in a year, so that thousands of years would be required for them 
to change their place in any striking degree, and hundreds of 
thousands to make a complete revolution around the heavens. 



g 2. PROPER MOTION OF THE SUN. 

A very interesting result of the proper motions of the 
stars is that our sun, considered as a star, has a consider- 
able proper motion of its own. By observations on a star, 
we really determine, not the proper motion of the star it- 
self, but the relative proper motion of the observer and 
the star — that is, the difference of their motions. Since 
the earth with the observer on it is carried along with the 
sun in space, his proper motion is the same as that of the 
sun, so that what observation gives us is the difference 
between the proper motion of the star and that of the sun. 
There is no way to determine absolutely how much of 
the apparent proper motion is due to the real motion of 
the star and how much to the real motion of the sun. If, 
however, we find that, on the average, there is a large pre- 
ponderance of proper motions in one direction, we may 
conclude that there is a real motion of the sun in an op- 
posite direction. This conclusion is reasonable, since it is 
more likely that the average of a great mass of stars is at 
rest than that the sun, which is only a single star, should 
be. Now, observation shows that this is really the case, 
and that the great mass of stars appear to be moving from 
the direction of the constellation Hercules and toward 



474 



ASTRONOMY. 



that of tlie constellation Argus.* A number of astrono- 
mers have investigated this motion with a view of deter- 
mining the exact point in the heavens toward which the 
sun is moving. Their results are shown in the following 
table : 



Right Ascension. 



Declination. 



Arjrelander 

O. Struve 

Lundahl 

Galloway 

Madler 

Airy and Dunkin 




50' N. 

36' N. 

14° 26' N. 

34° 23' N. 

54' N. 

58' N. 



28° 
37 c 



39 : 



It will be perceived that there is some discordance aris- 
ing from the diverse characters of the motions to be in- 
vestigated. Yet, if we lay these different points down on 
a map of the stars, we shall find that they all fall in the 
constellation Hercules. The amount of the motion is such 
that if the sun were viewed at right angles to the direction 
of motion from an average star of the first magnitude, it 
would appear to move about one third of a second per 
year. 

§ 3. DISTANCES OF THE FIXED STARS. 

The problem of the distance of the stars has always 
been one of the greatest interest on account of its involv- 
ing the question of the extent of the visible universe. 
The ancient astronomers supposed all the fixed stars to be 
situated at a short distance outside of the orbit of the planet 
Saturn , then the outermost known planet. The idea was 
prevalent that Nature would not waste space by leaving a 
great region beyond Saturn entirely empty. 

When Copernicus announced the theory that the sun 
was at rest and the earth in motion around it, the prob- 
lem of the distance of the stars acquired a new interest. 

* This was discovered by Sir William Herschel in 1783. 



DISTANCES OF THE STABS. 475 

It was evident that if the earth described an annual orbit, 
then the stars would appear in the course of a year to os- 
cillate back and forth in corresponding orbits, unless they 
were so immensely distant that these oscillations were too 
small to be seen. Now, the apparent oscillation of Saturn 
produced in this way was described in Part I., and shown 
to amount to some 6° on each side of the mean position. 
These oscillations were, in fact, those which the ancients 
represented by the motion of the planet around a small 
epicycle. But no such oscillation had ever been detected 
in a fixed star. This fact seemed to present an almost 
insuperable difficulty in the reception of the Copernican 
system. This was probably the reason why Tycho Bra he 
was led to reject the system. Very naturally, therefore, 
as the instruments of observation were from time to time 
improved, this apparent annual oscillation of the stars was 
ardently sought for. When, about the year 1704, 
Boemer thought he had detected it, he published his ob- 
servations in a dissertation entitled ^Copernicus Trium- 
phans." "A similar attempt, made by Hooke of England, 
was entitled " An Attempt to Prove the Motion of the 
Earth." 

This problem is identical with that of the annual paral- 
lax of the fixed stars, which has been already described in 
the concluding section of our opening chapter. This 
parallax of a heavenly body is the angle which the mean 
distance of the earth from the sun subtends when seen 
from the body. The distance of the body from the sun is 
inversely as the parallax (nearly). Thus the mean distance 
of Saturn being 9.5, its annual parallax exceeds 6°, while 
that of Neptune, which is three times as far, is about 2°. 
It was very evident, without telescopic observation, that 
the stars could not have a parallax of one half a degree. 
They must therefore be at least twelve times as far as 
Saturn if the Copernican system were true. 

When the telescope was applied to measurement, a con- 
tinually increasing accuracy began to be gained by the 



476 ASTRONOMY. 

improvement of the instruments. Yet for several genera- 
tions the parallax of the fixed stars eluded measurement. 
Yerj often indeed did observers think they had detected 
a parallax in some of the brighter stars, but their succes- 
sors, on repeating their measures with better instruments, 
and investigating their methods anew, found their con- 
clusions erroneous. Early in the present century it be- 
came certain that even the brighter stars had not, in gen- 
eral, a parallax as great as 1", and thus it became certain 
that they must lie at a greater distance than 200,000 times 
that which separates the earth from the sun. 

Success in actually measuring the parallax of the stars 
was at length obtained almost simultaneously by two as- 
tronomers, Bessel of Konigsberg, and Stkuve of Dorpat. 
Bessel selected for his star to be observed 61 Cygni, and 
commenced his observations on it in August, 1S3T. The 
result of two or three years of observation was that this 
star had a parallax of 0".35, or about one-third of a sec- 
ond. This would make its distance from the sun nearly 
600,000 astronomical units. The reality of this paral- 
lax has been well established by subsequent investigators, 
only it has been shown to be a little larger, and therefore 
the star a little nearer than Bessel supposed. The most 
probable parallax is now found to be 0".51, corresponding 
to a distance of 400,000 radii of the earth's orbit. 

The star selected by Struve ior the measure of parallax was the 
bright one, a Lyra. His observations were made between Novem- 
ber, 1835, and August, 1838. He first deduced a parallax of 0".25. 
Subsequent observers have reduced this parallax to 0-20, corre- 
sponding to a distance of about 1,000,000 astronomical units. 

Shortly after this, it was found by Hendersox, of England, As- 
tronomer Royal for the Cape of Good Hope, that the star a Centauri 
had a still larger parallax of about 1 ". This is the largest parallax 
now known in the case of any fixed star, so that a Centauri is, be- 
yond all reasonable doubt, the nearest fixed star. Yet its distance 
is more than 200,000 astronomical units, or thirty millions of mill- 
ions of kilometres. Light, which passes from the sun to the earth 
in 8 minutes, would require 3£ years to reach us from a Centauri. 

Two methods of determining parallax have been applied in as- 
tronomy. The parallax found by one of these methods is known as 
absolute, that by the other as relative parallax. In determining the 



DISTANCES OF THE STARS. 



47? 



absolute parallax, the observer finds the polar distance of the star 
as often as possible through a period of one or more years with a 
meridian circle, and then, by a discussion of all his observations, 
concludes what is the magnitude of the oscillation due to parallax. 
The difficulty in applying this method is that the refraction of the 
air and the state of the instrument are subject to changes arising 
from varying temperature, so that the observations are always un- 
certain by an amount which is important in such delicate work. 

In determining the relative parallax, the astronomer selects two 
stars in the same field of view of his telescope, one of which is 
many times more distant than the other. It is possible to judge 
with a high degree of probability which star is the more distant, 
from the magnitudes and proper motions of the two objects. It is 
assumed that a star which is either very bright or has a large pro- 
per motion is many times nearer to us than the extremely faint 
stars which may be nearly always seen around it. The effect of 
parallax will then be to change the apparent position of the bright 
star among the small stars around it in the course of a year. This 
change admits of being measured with great precision by the mi- 
crometer of the equatorial, and thus the relative parallax maybe 
determined. 

It is true that this relative parallax is really not the absolute par- 
allax of either body, but the difference of their parallaxes. So we 
must necessarily suppose that the parallax of the smaller and more 
distant object is zero. It is by this method of relative parallax 
that the great majority of determinations have been made. 

The distances of the stars are sometimes expressed by 
the time required for light to pass from them to our sys- 
tem. The velocity of light is, it will be remembered, 
about 300,000 kilometres per second, or such as to pass 
from the sun to the earth in 8 minutes 18 seconds. 

The time required for light to reach the earth from 
some of the stars, of which the parallax has been measured, 
is as follows : 



Star. 


Years. 


Star. 


Years. 


ex Centauri 


3-5 

6-7 

6-3 

6-9 

9-4 

10-5 

11-9 

13-1 

16-7 

17-9 


70 Ophiuchi 

l Urscv Major is 

Arcturus 

y Draconis 

1830 Groombridge. 

Polaris 


19-1 


61 Cygni 


24-3 


21,185 Lalando 

6 Centauri 

(i Cassiopeim 

34 Groombridge. . . . 

21,258 Lalande 

17,415 Oeltzen 

Sirius 


25-4 
35-1 
35-9 
42-4 


3077 Bradley 

85 Pegask ........ 

ex Auriga' 


46-1 
64-5 

70-1 


a. Lyrce 


a Draconis 


129-1 







CHAPTER VII. 

CONSTRUCTION OP THE HEAVENS. 

The visible universe, as revealed to us by the telescope, 
is a collection of many millions of stars and of several 
thousand nebulae. It is sometimes called the stellar or 
sidereal system, and sometimes, as already remarked, the 
stellar universe. The most far-reaching question with 
which astronomy has to deal is that of the form and mag- 
nitude of this system, and the arrangement of the stars 
which compose it. 

It was once supposed that the stars were arranged on 
the same general plan as the bodies of the solar system, 
being divided up into great numbers of groups or clus- 
ters, while all the stars of each group revolved in regular 
orbits round the centre of the group. All the groups were 
supposed to revolve around some great common centre, 
which was therefore the centre of the visible universe. 

But there is no proof that this view is correct. The 
only astronomer of the present century who held any such 
doctrine was Maedler. He thought that the centre of 
motion of all the stars was in the Pleiades, but no other 
astronomer shared his views. We have already seen that 
a great many stars are collected into clusters, but there is 
no evidence that the stars of these clusters revolve in 
regular orbits, or that the clusters themselves have any 
regular motion arcund a common centre. Besides, the 
large majority of stars visible with the telescope do not 
appear to be grouped into clusters at all. 



STRUCTURE OF THE HEAVENS. 



479 



The first astronomer to make a careful study of the 
arrangement of the stars with a view to learn the structure 
of the heavens was Sir William Herschel. He published 
in the Philosophical Transactions several memoirs on the 
construction of the heavens and the arrangement of the 
stars, which have become justly celebrated. We shall 
therefore begin with an account of Herschel's methods 
and results. 

Hekschel's method of study was founded on a mode of 
observation which he called star-gauging. It consisted in 
pointing a powerful telescope toward various parts of the 
heavens and ascertaining by actual count how thick the 
stars were in each region. His 20-foot reflector was pro- 
vided with such an eye-piece that, in looking into it, he 
would see a portion of the heavens about 15' in diameter. 
A circle of this size on the celestial sphere has about one 
quarter the apparent surface of the sun, or of the full 
moon. On pointing the telescope in any direction, a 
greater or less number of stars were nearly always visible. 
These were counted, and the direction in which the tele- 
scope pointed was noted. Gauges of this kind were made 
in all parts of the sky at which he could point his instru- 
ment, and the results were tabulated in the order of right 
ascension. 

The following is an extract from the gauges, and gives 
the average number of stars in each field at the points 
noted in right ascension and north polar distance : 





N. P. D. 




N. P. D. 


R.A. 


92° to 94° 


R.A. 


78° to 80° 




No. of Stars. 




No. of Stars. 


h. m. 




h. m. 




15 10 


9-4 


11 6 


3-1 


15 22 


10-6 


12 31 


3-4 


15 47 


10-6 


12 44 


4-6 


16 8 


12-1 


12 49 


3-9 


16 25 


13-6 


13 5 


3-8 


16 37 


18-6 


14 30 


3-6 



480 ASTRONOMY. 

In this small table, it is plain that a different law of 
clustering or of distribution obtains in the two regions. 
Such differences are still more marked if we compare the 
extreme cases found by Herschel, as R. A. = 19 h 41 m , 
N. P. D. = 74° 33', number of stars per field ; 588, 
and E. A. = 16 h 10 m , N. P. D., 113° 4', number of 
stars = 1-1. 

The number of these stars in certain portions is very 
great. For example, in the Milky Way, near Orion, six 
fields of view promiscuously taken gave 110, 60, TO, 90, 
70, and 74 stars each, or a mean of 79 stars per field. 
The most vacant space in this neighborhood gave 60 stars. 
So that as Herschel' s sweeps were two degrees wide in 
declination, in one hour (15°) there would pass through 
the field of his telescope 40,000 or more stars. In some 
of the sweeps this number was as great as 116,000 stars, 
in a quarter of an hour. 

On applying this telescope to the Milky Way, Her- 
schel supposed at the time that it completely resolved the 
whole whitish appearance into small stars. This conclu- 
sion he subsequently modiiied. He says : 

" It is very probable that the great stratum called the Milky "Way 
is that in which the sun is placed, though perhaps not in the very 
centre of its thickness. 

" We gather this from the appearance of the Galaxy, -which 
seems to encompass the whole heavens, as it certainly must do if 
the sun is within it. For, suppose a number of stars arranged be- 
tween two parallel planes, indefinitely extended every way, but at 
a given considerable distance from each other, and calling this a 
sidereal stratum, an eye placed somewhere within it will see all 
the stars in the direction of the planes of the stratum projected into 
a great circle, which will appear lucid on account of the accumu- 
lation of the stars, while the rest of the heavens, at the sides, will 
only seem to be scattered over with constellations, more or less 
crowded, according to the distance of the planes, or number of 
stars contained in the thickness or sides of the stratum." 

Thus in Herschel's figure an eye at £ within the stratum al> 
will see the stars in the direction of its length a 5, or height c d, 
with all those in the intermediate situations, projecced into the 
lucid circle A CBD, while those in the sides mv, nw, will be seen 
scattered over the remaining part of the heavens M VNW. 



STRUCTURE OF THE HEAVEN8. 



481 



" If the eye were placed somewhere without the stratum, at no 
very great distance, the appearance of the stars within it would 
assume the form of one of the smaller circles of the sphere, which 



^ast-V 









*Mfi • ♦J 



> iTf* 



f + « »»?»* * 

* i *~ ' .7 . « * 



* i * J. !*** " 



Mil ' * * 











Fig. 135. — herschel's theory op the stellar system. 

would be more or less contracted according to the distance of the 
eye ; and if this distance were exceedingly increased, the whole 
stratum might at last be drawn together into a lucid spot of any 



482 ASTRONOMY. 

shape, according to the length, breadth, and height of the stra- 
tum. 

1 * Suppose that a smaller stratum p q should branch out from 
the former in a certain direction, and that it also is contained 
between two parallel planes, so that the eye is contained within 
the great stratum somewhere before the separation, and not far 
from the place where the strata are still united. Then this second 
stratum will not be projected into a bright circle like the former, 
but it will be seen as a lucid branch proceeding from the first, and 
returning into it again at a distance less than a semicircle. 

"In the figure the stars in the small stratum pq will be pro- 
jected into a bright arc P R R P, which, after its separation from 
the circle C B D, unites with it again at P. 

"If the bounding surfaces are not parallel planes, but irregularly 
curved surfaces, analogous appearances must result." 

The Milky Way, as we see it, presents the aspect which 
has been just accounted for, in its general appearance of a 
girdle around the heavens and in its bifurcation at a cer- 
tain point, and Herschel's explanation of this appear- 
ance, as just given, has never been seriously questioned. 
One doubtful point remains: are the stars in Fig. 135 
scattered all through the space S — ahp dl or are they 
near its bounding planes, or clustered in any way within 
this space so as to produce the same result to the eye as if 
uniformly distributed ? 

Herschel assumed that they were nearly equably ar- 
ranged all through the space in question. He only exam- 
ined one other arrangement — viz., that of a ring of stars 
surrounding the sun, and he pronounced against such an 
arrangement, for the reason that there is absolutely noth- 
ing in the size or brilliancy of the sun to cause us to sup- 
pose it to be the centre of such a gigantic system. No 
reason except its importance to us personally can be alleged 
for such a supposition. By the assumptions of Fig. 135,. 
each star will have its own appearance of a galaxy or milky 
way, which will vary according to the situation of the star. 

Such an explanation will account for the general appear- 
ances of the Milky Way and of the rest of the sky, sup- 
posing the stars equally or nearly equally distributed in 
space. On this supposition, the system must be deeper 



STRUCTURE OF THE HEAVENS. 



48S 



where the stars appear more numerous. The same evi- 
dence can be strikingly presented in another way so as to 
include the results of the southern gauges of iSir John 
Herschel. The Galaxy, or Milky Way, being nearly a 
great circle of the sphere, we may compute the position 
of its north or south pole; and as the position of our own 
polar points can evidently have no relation to the stellar 
universe, we express the position of the gauges in galactic 
polar distance, north or south. By subtracting these 
polar distances from 90°, we shall have the distance of each 
gauge from the central plane of the Galaxy itself, the stars 
near 90° of polar distance being within the Galaxy. The 
average number of stars per field of 15' for each zone of 
15° of galactic polar distance has be^n tabulated by Struvb 
and Herschel as follows: 



Zones of Galactic 


Average Number 
of Stard per 


Zones of 


Average Number 


North Polar 


Galactic South Polar 


of Stars per 


Distance. 


Field of 15'. 


Distance. 


Field of 15'. 


0° to 15° 


4-32 


0° to 15° 


6-05 


15° to 30° 


5-42 


15° to 30° 


6-62 


30° to 45° 


8-21 


30° to 45° 


9-08 


45° to 60° 


13-61 


45° to 60° 


13-49 


60° to 75° 


24-09 


60° to 75° 


26-29 


75° to 90° 


53-43 


75° to 90° 


59-06 



This table clearly shows that the superficial distribution 
of stars from the first to the fifteenth magnitudes over the 
apparent celestial sphere is such that the vast majority of 
them are in that zone of 30° wide, which includes the 
Milky Way. Other independent researches have shown 
that the fainter lucid stars, considered alone, are also dis- 
tributed in greater number in this zone. 



Herschel endeavored, in his early memoirs, to find the physical 
explanation of this inequality of distribution in the theory of the 
universe exemplified in Fig. 136, which was based on the funda- 
mental assumption that, on the whole, the htars were nearly equably 
distributed in space. 



484 ASTRONOMY. 

If they were so distributed, then the number of stars visible in 
any gauge would show the thickness of the stellar system in the 
direction iu which the telescope was pointed. At each pointing, 
the field of view of the instrument includes all the visible stars sit- 
uated within a cone, having its vertex at the observer's eye, and its 
base at the very limits of the system, the angle of the cone (at the 
eye) being 15' 4". Then the cubes of the perpendiculars let fall 
from the eye on the plane of the bases of the various visual cones 
are proportional to the solid contents of the cones themselves, or, as 
the stars are supposed equally scattered within all the cones, the 
cube roots of the numbers of stars in each of the fields express the 
relative lengths of the perpendiculars. A section of the sidereal sys- 
tem along any great circle can thus be constructed as in the figure, 
which is copied from Herschel. 

The solar system is supposed to be at the dot within the mass of 
stars. From this point lines are drawn along the directions in 
which the gauging telescope was pointed. On these lines are laid 
off lengths proportional to the cube roots of the number of stars in 
each gauge. 




Fig. 136. — arrangement of the stars on the hypothesis op 
equable distribution. 

The irregular line joining the terminal points is approximately 
the bounding curve of the stellar system in the great circle chosen. 
Within this line the space is nearly uniformly filled with stars. 
"Without it is empty space. A similar section can be constructed in 
any other great circle, and a combination of all such would give a 
representation of the shape of our stellar system. The more numer- 
ous and careful the observations, the more elaborate the represen- 
tation, and the 863 gauges of Herschel are sufficient to mark out 
with great precision the main features of the Milky Way, and even 
to indicate some of its chief irregularities. This figure may be 
compared with Fig. 135. 

On the fundamental assumption of Herschel (equable distribu- 
tion), no other conclusions can be drawn from his statistics but 
that drawn by him. 

This assumption he subsequently modified in some degree, and 
was led to regard his gauges as indicating not so much the depth 
of the system in any direction as the clustering power or tendency 
of the stars in those special regions. It is clear that if in any 



STRUCTURE OF THE HEAVENS. 485 

given part of the sky, where, on the average, there are 10 stars 
(say) to a field, we should find a certain small portion of 100 or 
more to a field, then, on Herschel's first hypothesis, rigorously in- 
terpreted, it would be necessary to suppose a spike-shaped protu- 
berance directed from the earth in order to explain the increased 
number of stars. If many such places could be found, then the 
probability is great that this explanation is wrong. We should 
more rationally suppose some real inequality of star distribution 
here. It is, in fact, in just such details that the system of Her- 
schel breaks down, and the careful examination which his system 
has received leads to the belief that it must be greatly modified to 
cover all the known facts, while it undoubtedly has, in the main, a 
strong basis. 

The stars are certainly not uniformly distributed, and any gen- 
eral theory of the sidereal system must take into account the varied 
tendency to aggregation in various parts of the sky. 

The curious convolutions of the Milky Way, observed at various 
parts of its course, seem inconsistent with the idea of very great 
depth of this stratum, and Mr. Proctor has pointed out that the 
circular forms of the two " coal-sacks" of the Southern Milky Way 
indicate that they are really globular, instead of being cylindric 
tunnels of great length, looking into space, with their axes directed 
toward the earth. If they are globular, then the depth of the 
Milky Way in their neighborhood cannot be greatly different from 
their diameters, which would indicate a much smaller depth than 
that assigned by Herschel. 

In 1817, Herschel published an important memoir on the same 
subject, iii which his first method was largely modified, though 
not abandoned entirely. Its fundamental principle was stated by 
him as follows : 

" It is evident that we cannot mean to affirm that the stars of the 
fifth, sixth, and seventh magnitudes are really smaller than those 
of the first, second, or third, and that we must ascribe the cause 
of the difference in the apparent magnitudes of the stars to a differ- 
ence in their relative distances from us. On account of the great 
number of stars in each class, we must also allow that the stars of 
each succeeding magnitude, beginning with the first, are, one with 
another, further from us than those of the magnitude immediately 
preceding. The relative magnitudes give only relative distances, 
and can afford no information as to the real distances at which the 
stars are placed. 

" A standard of reference for the arrangement of the 'stars may 
be had by comparing their distribution to a certain properly mod- 
ified equality of scattering. The equality which I propose does not 
require that the stars should be at equal distances from each other, 
nor is it necessary that all those of the same nominal magnitude 
should be equally distant from us." 

It consists of allotting a certain equal portion of space to every 
star, so that, on the whole, each equal portion of space within the 
stellar system contains an equal number of stars. 



486 



ASTRONOMY. 



The space about each star can be considered spherical. Sup* 
pose such a sphere io surround our own sun, its radius will not 

differ greatly from the 
distance of the nearest 
fixed star, and this is 
taken as the unit of 
distance. 

Suppose a series of 
larger spheres, all 
drawn around our sun 
as a centre, and having 
the radii 3, 5, 7, 9, 
etc. The contents of 
the spheres being as 
the cubes of their 
diameters, the first 
sphere will have 3x3 
x 3 = 27 times the 
volume of the unit 
sphere, and will there- 
fore be large enough 
to contain 27 stars ; 
the second will have 
125 times the volume, 
and will therefore con- 
tain 125 stars, and so 
with the successive 
spheres. The figure 
shows a section of 
portions of these 
spheres up to that 
with radius 11. Above 
the centre are given 
the various orders of 
stars which are situ- 
ated between the sev- 
eral spheres, while 
in the corresponding 
spaces below the cen- 
tre are given the num- 
ber of stars which the region is large enough to contain ; for in- 
stance, the sphere of radius 7 has room for 343 stars, but of this 
space 125 parts belong to the spheres inside of it : there is, there- 
fore, room for 218 stars between the spheres of radii 5 and 7. 

Herschel designates the several distances of these layers of 
stars as orders ; the stars between spheres 1 and 3 are of the first 
order of distance, those between 3 and 5 of the second order, and 
so on. Comparing the room for stars between the several spheres 
with the number of stars of the several magnitudes, he found the 
result to be as follows : 




Fig. 137. — orders of distance op stars. 



STRUCTURE OF THE HEAVENS. 



487 



Order of Distance. 


Number of Stars 
there is Room for. 


Magnitude. 


Number of Stars 
of that Magnitude. 


1 

2 


26 

98 
218 
386 
602 
866 
1,178 
1,538 


1 

2 
3 

4 
5 
6 

7 


17 
57 


3 


206 


4 


454 


5 


1,161 
6,103 


6 


7 


6,146 


8 







The result of this comparison is, that, if the order of magnitudes 
could indicate the distance of the stars, it would denote at first a 
gradual and afterward a very abrupt condensation of them. 

If, on the ordinary scale of magnitudes, we assume the brightness 
of any star to be inversely proportional to the square of its dis- 
tance, it leads to a scale of distance different from that adopted by 
Herschel, so that a sixth- magnitude star on the common scale 
would be about of the eighth order of distance according to this 
scheme — that is, we must remove a star of the first magnitude to 
eight times its actual distance to make it shine like a star of the 
sixth magnitude. 

On the scheme here laid down, Herschel subseauently assigned 
the order of distance of various objects, mostly star-clusters, and 
his estimates of these distances are still quoted. They rest on the 
fundamental hypothesis which has been explained, and the error 
in the assumption of equal brilliancy for all stars, affects these esti- 
mates. It is perhaps most probable that the hypothesis, of equal 
brilliancy for all stars is still more erroneous than the hypothesis 
of equal distribution, and it may well be that there is a very large 
range indeed in the actual dimensions and in the iutrinsic brilliarjcy 
of stars at the same order of distance from us, so that the tenth- 
magnitude stars, for example, may be scattered throughout the 
spheres, which Herschel would assign to the seventh, eighth, 
ninth, tenth, eleventh, twelfth, and thirteenth magnitudes. 

Since the time of Herschel, one of the most eminent of the as- 
tronomers who have investigated this subject is Struve the elder, 
formerly director of the Pulkowa Observatory. His researches 
were founded mainly on the numbers of stars of the several magni- 
tudes found by Bessel in a zone thirty degrees wide extending all 
around the heavens, 15° on each side of the equator. With these 
he combined the gauges of Sir William Herschel. The hypothesis 
on which he based his theory was similar to that employed by 
Herschel in his later researches, in so far that he supposed the 
magnitude of the stars to furnish, on the average, a measure of 
their relative distances. Supposing, after Herschel, a number of 
concentric spheres to be drawn around the sun as a centre, the suc- 
cessive spaces between which corresponded to stars of the several 



488 



ASTRONOMY. 



magnitudes, he found that the further out he went, the more the 
stars were condensed in and near the Milky Way. This conclusion 
may be drawn at once from the fact we have already mentioned, 
that the smaller the stars, the more they are condensed in the re- 
gion of the Galaxy. Struve found that if we take only the stars 
plainly visible to the naked eye — that is, those down to the fifth 
magnitude — they are no thicker in the Milky Way than in other 
parts of the heavens. But those of the sixth magnitude are a 
little thicker in that region, those of the seventh yet thicker, and 
so on, the inequality of distribution becoming constantly greater as 
the telescopic power is increased. 

From all this, Struve concluded that the stellar system might 
be considered as composed of layers of stars of various densities, all 
parallel to the plane of the Milky Way. The stars are thickest in and 
near the central layer, which he conceives to be spread out as a wide, 
thin sheet of stars. Our sun is situated near the middle of this 
layer. As we pass out of this layer, on either side we find the 
stars constantly growing thinner and thinner, but we do not reach 
any distinct boundary. As, if we could rise in the atmosphere, we 
should find the air constantly growing thinner, but at so gradual a 
rate of progress that we could hardly say where it terminated ; so, 
on Struve's view, would it be with the stellar system, if we could 
mount up in a direction perpendicular to the Milky Way. Struve 
gives the following table of the thickness of the stars on each side 
of the principal plane, the unit of distance being that of the ex- 
treme distance to which Herschel's telescope conld penetrate : 









^fean Distance 


Distance from Pr 


ncipal Plane. 


Density. 


between Neighbor- 
ing Stars. 


In the principal pla 
0-05 from principa 


ne 


1-0000 


1-000 


plane 


0-48568 


1-272 


0-10 


11 


1 


0-33288 


1-458 


0-20 


« 


' 


0-23895 


1-611 


0-30 


< 


i 


0-17980 


1-772 


0-40 


< 


< 


0-13021 


1-973 


0-50 


tt 


it 


0-08646 


2-261 


0-60 


c 


< 


0-05510 


2-628 


0-70 


« 


< 


0-03079 


3-190 


0-80 


€ i 


« 


0-01414 


4-131 


0-866 


t 


0-00532 


5-729 



This condensation of the stars near the central plane and the 
gradual thinning-out on each side of it are only designed to be the 
expression of the general or average distribution of those bodies. 
The probability is that even in the central plane the stars are many 
times as thick in some regions as in others, and that, as we leave the 
plane, the thinning-out would be found to proceed at very different 
rates in different regions. That there may be a gradual thinning-out 



STRUCTURE OF THE HEAVENS. 489 

cannot be denied ; but Struve's attempt to form a table of it is open 
to the serious objection that, like Herschel, he supposed the differ- 
ences between the magnitudes of the stars to arise entirely from 
their different distances from us. Although where the scattering 
of the stars is nearly uniform, this supposition may not lead us into 
serious error, the case will be entirely different where we have to 
deal with irregular masses of stars, and especially where our tele- 
scopes penetrate to the boundary of the stellar system. In the 
latter case we cannot possibly distinguish between small stars lying 
within the boundary and larger ones scattered outside of it, and 
Struve's gradual thinning-out of the stars may be entirely ac- 
counted for by great diversities in the absolute brightness of the 
stars. 

Distribution of Stars. — The brightness B of any star, as seen 
from the earth, depends upon its surface S, the intensity of its light 
per unit of surface, i, and its distance D, so that its brightness can be 
expressed thus : 

S xi 



B = 



for another star : B' 

and 



Z> 2 

8 x i 



B S>i Z>' 2 



B' s' • e D 2 

Now this ratio of the brightness B 4- B' is the only fact we usually 
know with regard to any two stars. D has been determined for 
only a few stars, and for these it varies between 200,000 and 2,000,000 
times the mean radius of the earth's orbit. #and i are not known for 
any star. There is, however, a probability that i does not vary greatly 
from star to star, as the great majority of stars are white in color (only 
some 700 red stars, for instance, are known out of the 300,000 which 
have been carefully examined). Among 476 double stars of Struve's 
list 295 were white, 63 being bluish, only one fourth, or 118, being 
yellow or red. 

If B is of the r^th mag. its light in terms of a first magnitude star 
is 6 n — » where 6 = 0-397, and if B' is of the rath mag., its light is 
<5 m — >, both expressed in terms of the light of a first magnitude star as 
unity (6° = 1). 

Therefore we may put B = 6 n — J , B / = d"*~ ', and we have 



8' -i'-IP 



In this general expression we seek the ratio -=— , and we have it 

expressed in terms of four unknown quantities. We must therefore 
make some supposition in regard to these. 

I. If all stars are of equal intrinsic brilliancy and of equal size, then 

Si = S' i\ and ** -»sa constant = — ™", 



490 ASTRONOMY, 

whence the relative distance of any two stars would be known on this 
hypothesis. 

II. Or, suppose the stars to be uniformly distributed in space, or the 
star-density to be equal in all directions. From this we can also 
obtain some notions of the relative distances of stars. 

Call D lt D 2 , D 3 D n the average distances of stars of the 

1, 2, 3, ..... ntW magnitudes. 

If K stars are situated within the sphere of radius 1, then the num- 
ber of stars (Q n ), situated within tlie sphere of radius D n , is 

Q H = K • {D n )\ 

since the cubic contents of spheres are as the cubes of their radii. 
Also 

Q»-i = K{Dn-ip, 
whence 

3 



S7Z\- V «._,- 



If we knew Q n and Q n — i, the number of stars contained in the 
spheres of radii D n and D n _ i, then the ratio of D n and D n - i would 
be known. We cannot know Q n , Q n _ lf etc., directly, but we may 
suppose these quantities to be proportional to the numbers of stars of 
the ?ith and (n — l)tli magnitudes found in an enumeration of all the 
stars in the heavens of these magnitudes, or, failing in these data, we 
may confine this enumeration to the northern hemisphere, where 
Littrow has counted the number of stars of each class in Argelan- 
DEr's Durchmusterung . As we have seen (p. 436) 

Q, = 19,699 and Q 6 = 77,794, 
whence 



£ = •-$- = !•• 



and this would lead us to infer that the stars of the 8th magnitude 
were distributed inside of a sphere whose radius was about 1-6 times 
that of the corresponding sphere for the 7th magnitude stars provided 
that, 1st, the stars in general are equally or about equally distributed, 
and, 2d, that on the whole the stars of the 8 .... n magnitudes are 

further away from us than those of the 7 (n — 1) magnitudes. 

We may have a kind of test of the truth of this hypothesis, and of 
the first employed, as follows : we had 



Dn _ I/ Qn 

Dn-X ~ y Qn-l' 



Also from the first hypothesis the briirhtness B n of a star of the nth, 
magnitude in terms of a first magnitude star = 1 was 

Bn •-= <5» - ». 

If here, again, we suppose the distance of a first magnitude star to 
be = 1 and of an Tith magnitude star D», then 



STRUCTURE OF THE HEAVENS. 491 

or 



*--.HW"". 



Also 



whence ^ = — • • 

Comparing the expression for — -- — , in the two cases, we have 

r ¥»-l VT \ Qn J 

If the value of d in this last expression comes near to the value which 
has been deduced lor it from direct photometric measures of the 
relative intensity of various classes of stars, viz., 6 = 0-40, then this 
will be so far an argument to show that a certain amount of credence 
may be given to both hypotheses I. and II. Taking the values of 
Qi and Q ti , we have 



'«•.•>-(-#&)'-* 



From the values of Q e ar *d Qu there results <5(e, 7) = 0-45. These, 
then, agree tolerably well with the independent photometric values 
for 6, and show that the equation 



1- 



-■ - Or)' 

of the nth magnitude with a 



gives the average distance of the stars of the nth magnitude with a 
certain approach to accuracy. For the stars from 1st to 8th magni- 
tude these distances are : 



1 to 1-9 magnitude 1-00 

2 to 2-9 " 1-54 

3 to 3-9 " 2-36 

4 to 4-9 " 3-64 

5 to 5-9 " 5-59 

6 to 6-9 «« 8-61 

7 to 7-9 «« 13-23 

8 to 8-9 " 20-35 

This presentation of the subject is essentially that of Prof Hugo 
Gylden. 



CHAPTER VIII. 

COSMOGONY. 

A theory of the operations by which the universe re- 
ceived its present form and arrangement is called Cosmog- 
ony, This subject does not treat of the origin of matter, 
but only with its transformations. 

Three systems of Cosmogony have prevailed among 
thinking men at different times. 

(1.) That the universe had no origin, but existed from 
eternity in the form in which we now see it. 

(2.) That it was created in its present shape in a 
moment, out of nothing. 

(3.) That it came into its present form through an ar- 
rangement of materials which were before ' ' without form 
and void." 

The last seems to be the idea which has most prevailed 
among thinking men, and it receives many striking con- 
firmations from the scientific discoveries of modern times. 
The latter seem to show beyond all reasonable doubt that 
the universe could not always have existed in its present 
form and under its present conditions ; that there was a time 
when the materials composing it were masses of glowing 
vapor, and that there will be a time when the present state 
of things will cease. The explanation of the processes 
through which this occurs is sometimes called the nebular 
hypothesis. It was first propounded by the philosophers 
Swedenborg, Kant, and Laplace, and although since 
greatly modified in detail, the views of these men have in 
the main been retained until the present time. 



COSMOGONY. 493 

We shall begin its consideration by a statement of the 
various facts which appear to show that the earth and 
planets, as well as the sun, were once a fiery mass. 

The first of these facts is the gradual but uniform in- 
crease of temperature as we descend into the interior of 
the earth. Wherever mines have been dug or wells sunk 
to a great depth, it is found that the temperature increases 
as we go downward at the rate of about one degree centi- 
grade to every 30 metres, or one degree Fahrenheit to 
every 50 feet. The rate differs in different places, but the 
general average is near this. The conclusion which we 
draw from this may not at first sight be obvious, because 
it may seem that the earth might always have shown this 
same increase of temperature. But there are several re- 
sults which a little thought will make clear, although their 
complete establishment requires the use of the higher 
mathematics. 

The first result is that the increase of temperature can- 
not be merely superficial, but must extend to a great 
depth, -probably even to the centre of the earth. If it did 
not so extend, the heat would have all been lost long ages 
ago by conduction to the interior and by radiation from 
the surface. It is certain that the earth has not received 
any great supply of heat from outside since the earliest 
geological ages, because such an accession of heat at the 
earth's surface would have destroyed all life, and even 
melted all the rocks. Therefore, whatever heat there is 
in the interior of the earth must have been there from be- 
fore the commencement of life on the globe, and remained 
through all geological ages. 

The interior of the earth being hotter than its surface, 
and hotter than the space around it, must be losing heat. 
We know by the most familiar observation that if any ob- 
ject is hot inside, the heat will work its way through to the 
surface by the process of conduction. Therefore, since the 
earth is a great deal hotter at the depth of 30 metres than 
it is at the surface, heat must be continually coming to the 



494 ASTRONOMY. 

surface. On reaching the surface, it must be radiated off 
into space, else the surface would have long ago become 
as hot as the interior. Moreover, this loss of heat must 
have been going on since the beginning, or, at least, since 
a time when the surface was as hot as the interior. Thus, if 
we reckon backward in time, we find that there must have 
been more and more heat in the earth the further back 
we go, so that we must finally reach back to a time when 
it was so hot as to be molten, and then again to a time 
when it was so hot as to be a mass of fiery vapor. 

The second fact is that we find the sun to be cooling off 
like the earth, only at an incomparably more rapid rate. 
The sun is constantly radiating heat into space, and, so far 
as we can ascertain, receiving none back again. A small 
portion of this heat reaches the earth, and on this portion 
depends the existence of life and motion on the earth's sur- 
face. The quantity of heat which strikes the earth is only 
about 2000 o 1 *) oooo °^ *hat which the sun radiates. This 
fraction expresses the ratio of , the apparent surface of the 
earth, as seen from the sun, to that of the whole celestial 
sphere. 

Since the sun is losing heat at this rate, it must have had 
more heat yesterday than it has to-day ; more two days ago 
than it had yesterday, and so on. Thus calculating back- 
ward, we find that the further we go back into time the 
hotter the sun must have been. Since we know that heat 
expands all bodies, it follows that the sun must have been 
larger in past ages than it is now, and we can trace back 
this increase in size without limit. Thus we are led to the 
conclusion that there must have been a time when the sun 
filled up the space now occupied by the planets, and must 
have been a very rare mass of glowing vapor. The plan- 
ets could not then have existed separately, but must have 
formed a part of this mass of vapor. The latter was there- 
fore the material out of which the solar system was 
formed. 

The same process may be continued into the future. 



COSMOGONY. 495 

Since the sun by its radiation is constantly losing heat, it 
must grow cooler and cooler as ages advance, and must 
finally radiate so little heat that life and motion can no 
longer exist on our globe. 

The third fact is that the revolutions of all the planets 
around the sun take place in the same direction and in 
nearly the same plane. We have here a similarity amongst 
the different bodies of the solar system, which must have 
had an adequate cause, and the only cause which has ever 
been assigned is found in the nebular hypothesis. This 
hypothesis supposes that the sun and planets were once 
a great mass of vapor, as large as the present solar system, 
revolving on its axis in the same plane in which the 
planets now revolve. 

The fourth fact is seen in the existence of nebulae. We 
have already stated that the spectroscope shows these bodies 
to be masses of glowing vapor. We thus actually see mat- 
ter in the celestial spaces under the very form in which 
the nebular hypothesis supposes the matter of our solar 
system to have once existed. Since these masses of vapor 
are so hot as to radiate light and heat through the immense 
distance which separates us from them, they must be grad- 
ually cooling off. This cooling must at length reach a 
point when they will cease to be vaporous and condense 
into objects like stars and planets. We know that every 
star in the heavens radiates heat as our sun does. In the 
case of the brio-liter stars the heat radiated has been made 
sensible in the foci of our telescopes by means of the thermo- 
multiplier. The general relation which we know to ex- 
ist between light and radiated heat shows that all the stars 
must, like the sun, be radiating heat into space. 

A fifth fact is afforded by the physical constitution of 
the planets Jupiter and Saturn. The telescopic examina- 
tion of these planets shows that changes on their surfaces 
are constantly going on with a rapidity and violence to 
which nothing on the surface of our earth can compare. 
Such operations can be kept up only through the agency of 



496 ASTRONOMY. 

heat or some equivalent form of energy. But at the dis- 
tance of Jupiter and Saturn the rajs of the sun are entirely 
insufficient to produce changes so violent. We are there- 
fore led to infer that Jupiter and Saturn must be hot 
bodies, and must therefore be cooling off like the sun, 
stars and earth. 

We are thus led to the general conclusion that, so far 
as our knowledge extends, nearly all the bodies of the 
universe are hot, and are cooling off by radiating their 
heat into space. Before the discovery of the " conserva- 
tion of energy," it was not known that this radiation in- 
volved the waste of a something which is necessarily limited 
in supply. But it is now known that heat, motion, and 
other forms of force are to a certain extent convertible into 
each other, and admit of being expressed as quantities of 
a general something which is called energy. We may de- 
fine the unit of energy in two or more ways : as the quan- 
tity which is required to raise a certain weight through a 
certain height at the surface of the earth, or to heat a given 
quantity of water to a certain temperature. However 
we express it, we know by the laws of matter that a given 
mass of matter can contain only a certain definite number 
of units of energy. When a mass of matter either gives 
off heat, or causes motion in other bodies, we know that 
its energy is being expended. Since the total quantity of 
energy which it contains is finite, the process of radiating 
heat must at length come to an end. 

It is sometimes supposed that this cooling off may be 
merely a temporary process, and that in time something 
may happen by which all the bodies of the universe will 
receive back again the heat which they have lost. This is 
founded upon the general idea of a compensating process in 
nature. As a special example of its application, some have 
supposed that the planets may ultimately fall into the sun, 
and thus generate so much heat as to reduce the sun once 
more to vapor. All these theories are in direct opposition 
to the well-established laws of heat, and can be justified 



COSMOGONY 497 

only by some generalization which shall be far wider than 
any that science has yet reached. Until we have snch a 
generalization, every such theory founded upon or consist- 
ent with the laws of nature is a necessary failure. All the 
heat that could be generated by a fall of all the planets into 
the sun would not produce any change in its constitution, 
and would only last a few years. The idea that the heat 
radiated by the sun and stars may in some way be collected 
and returned to them by the mere operation of natural laws 
is equally untenable. It is a fundamental principle of the 
laws of heat that a warm body can never absorb more 
heat from a cool one than the latter absorbs from it, and 
that a body can never grow warm in a space cooler than 
itself. All differences of temperature tend to equalize 
themselves, and the only state of things to which the uni- 
verse can tend, under its present laws, is one in which all 
space and all the bodies contained in space are at a uniform 
temperature, and then all motion and change of tempera- 
ture, and hence the conditions of vitality, must cease. And 
then all such life as ours must cease also unless sustained 
by entirely new methods. 

The general result drawn from all these laws and facts 
is, that there was once a time when all the bodies of the 
universe formed either a single mass or a number of masses 
of fiery vapor, having slight motions in various parts, and 
different degrees of density in different regions. A grad- 
ual condensation around the centres of greatest density then 
went on in consequence of the cooling and the mutual at- 
traction of the parts, and thus arose a great number of 
nebulous masses. One of these masses formed the ma- 
terial out of which the sun and planets are supposed to 
have been formed. It was probably at first nearly glob- 
ular, of nearly equal density throughout, and endowed 
with a very slow rotation in the direction in which the 
planets now move. As it cooled off, it grew smaller and 
smaller, and its velocity of rotation increased in rapidity by 
virtue of a well-established law of mechanics, known as 



498 ASTRONOMY. 

that of the conservation of areas. According to this law, 
whenever a system of particles of any kind whatever, which 
is rotating around an axis, changes its form or arrangement 
by virtue of the mutual attractions of its parts among them- 
selves, the sum of all the areas described by each particle 
around the centre of rotation in any unit of time remains 
constant. This sum is called the areolar velocity. 

If the diameter of the mass is reduced to one half, sup- 
posing it to remain spherical, the area of any plane section 
through its centre will be reduced to one fourth, because 
areas are proportional to the squares of the diameters. 
In order that the areolar velocity may then be the same 
as before, the mass must rotate four times as fast. The 
rotating mass we have described must have had an axis 
around which it rotated, and therefore an equator defined 
as being everywhere 90° from this axis. In consequence 
of the increase in the velocity of rotation, the centrifugal 
force would also be increased as the mass grew smaller. 
This force varies as the radius of the circle described by 
the particle multiplied by the square of the angular velocity. 
Hence when the masses, being reduced to half the radius, 
rotate four times as fast, the centrifugal force at the equa- 
tor would be increased ^ X 4 2 , or eight times. The gravi- 
tation of the mass at the surface, being inversely as the 
square of the distance from the centre, or of the radius, 
would be increased four times. Therefore as the masses 
continue to contract, the centrifugal force increases at a 
more rapid rate than the central attraction. A time would 
therefore come when they would balance each other at the 
equator of the mass. The mass would then cease to con- 
tract at the equator, but at the poles there would be no 
centrifugal force, and the gravitation of the mass would 
grow stronger and stronger. In consequence the mass would 
at length assume the form of a lens or disk very thin in pro- 
portion to its extent. The denser portions of this lens 
would gradually be drawn toward the centre, and there 
more or less solidified by the process of cooling. A point 



COSMOGONY. 499 

would at length be reached, when solid particles would begin 
to be formed throughout the whole disk. These would grad« 
ually condense around each other and form a single planet, or 
they might break up into small masses and form a group of 
planets. As the motion of rotation would not be altered 
by these processes of condensation, these planets would all 
be rotating around the central part of the mass, which is 
supposed to have condensed into the sun. 

It is supposed that at first these planetary masses, being 
very hot, were composed of a central mass of those sub- 
stances which condensed at a very high temperature, sur- 
rounded by the vapors of those substances which were 
more volatile. We know, for instance, that it takes a much 
higher temperature to reduce lime and platinum to vapor 
than it does to reduce iron, zinc, or magnesium. There- 
fore, in the original planets, the limes and earths would 
condense first, while many other metals would still be in a 
state of vapor. The planetary masses would each be 
affected by a rotation increasing in rapidity as they grew 
smaller, and would at length form masses of melted metals 
and vapors in the same way as the larger mass out of which 
the sun and planets were formed. These masses would 
then condense into a planet, with satellites revolving 
around it, just as the original mass condensed into sun and 
planets. 

At first the planets would be so hot as to be in a molten 
condition, each of them probably shining like the sun. 
They would, however, slowly cool off by the radiation of 
heat from their surfaces. So long as they remained liquid, 
the surface, as fast as it grew cool, would sink into the in- 
terior on account of its greater specific gravity, and its 
place would be taken by hotter material rising from the 
interior to the surface, there to cool off in its turn. There 
would, in fact, be a motion something like that which occurs 
when a pot of cold water is set upon the fire to boil. 
Whenever a mass of water at the bottom of the pot is 
heated, it rises to the surface, and the cool water moves 



500 ASTRONOMY. 

down to take its place. Thus, on the whole, so long as 
the planet remained liquid, it would cool off equally 
throughout its whole mass, owing to the constant motion 
from the centre to the circumference and back again. A 
time would at length arrive w T hen many of the earths and 
metals would begin to solidify. At first the solid particles 
would be carried up and down with the liquid. A time 
would finally arrive when they would become so large 
and numerous, and the liquid part of the general mass 
become so viscid, that the motion would be obstructed. 
The planet would then begin to solidify. Two views 
have been entertained respecting the process of solidifica- 
tion. 

According to one view, the whole surface of the planet 
would solidify into a continuous crust, as ice forms over a 
pond in cold weather, while the interior was still in a 
molten state. The interior liquid could then no longer 
come to the surface to cool off, and could lose no heat 
except what was conducted through this crust. Heuce 
the subsequent cooling would be much slower, and the 
globe would long remain a mass of lava, covered over by 
a comparatively thin solid crust like that on which we 
live. 

The other view is that, when the cooling attained a cer- 
tain stage, the central portion of the globe would be 
solidified by the enormous pressure of the superincumbent 
portions, while the exterior was still fluid, and that thus 
the solidification would take place from the centre out- 
ward. 

It is still an unsettled question whether the earth is now 
solid to its centre, or whether it is a ^reat globe of molten 
matter with a comparatively thin crust. Astronomers and 
physicists incline to the former view ; geologists to the 
latter one. Whichever view may be correct, it appears 
certain that there are great lakes of lava in the interior 
from which volcanoes are fed. 

It must be understood that the nebular hypothesis, as 



COSMOGONY. 501 

we have explained it, is not a perfectly established scien- 
tific theory, but only a philosophical conclusion founded 
on the widest study of nature, and pointed to by many 
otherwise disconnected facts. The widest generalization 
associated, with it is that, so far as we can see, the universe 
is not self-sustaining, but is a kind of organism which, like 
all other organisms we know of, must come to an end in 
consequence of those very laws of action which keep it 
going. It must have had a beginning within a certain 
number of years which we cannot yet calculate with cer- 
tainty, but which cannot much exceed 20,000,000, and it 
must end in a chaos of cold, dead globes at a calculable 
time in the future, when the sun and stars shall have 
radiated away all their heat, unless it is re-created by the 
action of forces of which we at present know nothing. 



FINIS. 



INDEX. 



gpgJT" This index is intended to point out the subjects treated in the 
work, and further, to give references to the pages where technical terms 
are denned or explained. 



Aberration-constant, values of, 
244. 

Aberration of a lens (chromatic), 
60. 

Aberration of a lens (spherical), 
61. 

Aberration of light, 238. 

Absolute parallax of stars denned, 
476. * 

Accelerating force denned, 140. 

Achromatic telescope described, 
60. 

Adams's work on perturbations of 
Uranus, 366. 

Adjustments of a transit instru 
ment are three ; for level, for 
collimation, and for azimuth, 77. 

Aerolites, 375. 

Airy's determination of the densi- 
ty of the earth, 193. 

Algol (variable star), 440. 

Altitude of a star denned, 25. 

Annular eclipses of the sun, 175. 

Autumnal equinox, 110. 

Apparent place of a star, 235. 

Apparent semi-diameter of a celes- 
tial body denned, 52. 

Apparent time, 260. 

Arago's catalogue of Aerolites, 
375. 

Arc converted into time, 32. 



Argelander's Durchmusterung, 
435. 

Argelander's manometry, 435. 

Aristarchus determines the solar 
parallax, 223. 

Aristarchus maintains the rota- 
tion of the earth, 14. 

Artificial horizon used with sex- 
tant on shore, 95. 

Aspects of the planets, 272 

Asten's, von, computation of 
orbit of Donati's comet, 409. 

Asteroids defined, 268. 

Asteroids, number of, 200 in 1879, 
341. 

Asteroids, their magnitudes, 341. 

Astronomical instruments (in gen- 
eral), 53. 

Astronomical units of length and 
mass, 214. 

Astronomy (defined), 1. 

Atmosphere of the moon, 331. 

Atmospheres of the planets, see 
Mercury, Venus, etc. 

Axis of the celestial sphere de- 
fined, 23. 

Axis of the earth defined, 25. 

Azimuth error of a transit instru- 
ment, 77. 

Baily's determination of the den- 
sity of the earth, 192. 



504 



INDEX. 



J 



Bayer's uranometry (1654), 420. 

Beer and Maedler's map of 
the moon, 333. 

Bessel's parallax of 61 Cygni 
(1837), 476. 

Bessel's work on the theory of 
Uranus, 366. 

Biela's comet, 404. 

Binary stars, 450. 

Binary stars, their orbits, 452. 

Bode's catalogue of stars, 435. 

Bode's law stated, 269. 

Bond's discovery of the dusky 
ring of Saturn, 1850, 356. 

Bond's observations of Donati's 
comet, 392. 

Bond's theory of the constitution 
of Saturn's rings, 359. 

Bouvard's tables of Uranus, 365. 

Bradley discovers aberration in 
1729, 240. 

Bradley's method of eye and ear 
observations (1750), 79. 

Brightness of all the stars of each 
magnitude, 439. 

Calendar, can it be improved ? 
261. 

Calendar of the French Republic, 
262. 

Calendars, how formed, 248. 

Callypus, period of, 252. 

Cassegrainian (reflecting) telescope, 
67. 

Cassini discovers four satellites of 
Saturn (1684-1671), 360. 

Cassini's value of the solar paral- 
lax, 9" -5, 226. 

Catalogues of stars, general ac- 
count, 434. 

Catalogues of stars, their arrange- 
ment, 265. 

Cavendish, experiment for deter- 
mining the density of the earth, 
192. 

Celestial mechanics defined, 3. 

Celestial sphere, 14, 41. 

Central eclipse of the sun, 177. 



Centre of gravity of the solar sys- 
tem, 272. 

Centrifugal force, a misnomer, 
210. 

Christie's determination of mo- 
tion of stars in line of sight, 471 . 

Chromatic aberration of a lens, 60. 

Chronograph used in transit ob- 
servations, ^9. 

Chronology, 245. 

Chronometers, 70. 

Clairatjt predicts the return of 
Halley's comet (1759), 397. 

Clarke's elements of the earth* 
202. 

Clocks, 70. 

Clusters of stars are often formed 
by central powers, 464. 

Coal-sacks in the milky way, 415, 
485. 

Coma of a comet, 388. 

Comets defined, 268. 

Comets formerly inspired terror, 
405-6. 

Comets, general account, 388. 

Comets' orbits, theory of, 400. 

Comets' tails, 388. 

Comets' tails, repulsive force, 395. 

Comets, their origin, 401. 

Comets,their physical constitution, 
393. 

Comets, their spectra, 393. 

Conjunction (of a planet with the 
sun) defined, 114. 

Collimation of a transit instru- 
ment, 77. 

Conjugate foci of a lens defined, 
65. 

Constellations, 414. 

Constellations, in particular, 422, 
et seq. 

Construction of the Heavens, 478. 

Co-ordinates of a star defined, 41. 

Copeland observes spectrum of 
new star of 1876, 445. 

Cornu's observations of spectrum 
of new star of 1876, 445. 



INDEX. 



505 



Cornu determines the velocity of 
light, 222. 

Correction of a clock defined, 72. 

Cosmical physics defined, 3. 

Cosmogony defined, 492. 

Corona, its spectrum, 305. 

Corona (the) is a solar appendage, 
302. 

Craters of the moon, 328. 

Day, how subdivided into hours, 
etc., 257. 

Days, mean solar, and solar, 259. 

Declination of a star defined, 20. 

Dispersive power of glass defined, 
61. 

Distance of the fixed stars, 412, 
474. 

Distribution of the stars, 489. 

Diurnal motion, 10. 

Diurnal paths of stars are circles 
12. 

Dominical letter, 255. 

Donati's comet (1858), 407. 

Double (and multiple) stars, 448. 

Double stars, their colors, 452. 

Earth (the), a sphere, 9. 

Earth (the) general account of, 188. 

Earth (the) is a point in compari- 
son with the distance of the fixed 
stars, 17. 

Earth (the) is isolated in space, 10. 

Earth's annual revolution, 98. 

Earth's atmosphere at least 100 
miles in height, 380. 

Earth's axis remains parallel to it- 
self during an annual revolution, 
109, 110. 

Earth's density, 188, 190. 

Earth's dimensions, 201. 

Earth's internal heat, 493. 

Earth's mass, 188. 

Earth's mass with various values 
of solar parallax (table), 230. 

Earth's motion of rotation proba- 
bly not uniform. 148. 

Earths' (the) relation to the heav- 
ens, 9. 



Earth's rotation maintained by 
Aristarchds and Timociiaris, 
and opposed by Ptolemy, 14. 

Earth's surface is gradually cool- 
ing, 493. 

Eccentrics devised by the ancients 
to account for the irregularities 
of planetary motions, 121. 

Eclipses of the moon, 170. 

Eclipses of the sun and moon, 168. 

Eclipses of the sun, explanation, 
172. 

Eclipses of the sun, physical phe- 
nomena, 297. 

Eclipses, their recurrence, 177. 

Ecliptic defined, 100. 

Ecliptic limits, 178. 

Elements of the orbits of the ma- 
jor planets, 276. 

Elliptic motion of a planet, its 
mathematical theory, 125. 

Elongation (of a planet) defined, 
114. 

Encke's comet, 409. 

Encke's value of the solar paral- 
lax, 8" -857, 226. 

Engelmann's photometric meas- 
ures of Jupiter's satellites, 350. 

Envelopes of a comet, 390. 

Epicycles, their theory, 119. 

Equation of time, 258. 

Equator (celestial) defined, 19, 24. 

Equatorial telescope, description 
of, 87. 

Equinoctial defined, 24. 

Equinoctial year, 207. 

Equinoxes, 104. 

Equinoxes ; how determined, 105. 

Evection, moon's 163. 

Eye-pieces of telescopes, 62. 

Eye (the naked) sees about 20Q0 
stars, 411, 414. 

Fabritius observes solar spots 
(1611), 288. 

Figure of the earth, 198. 

Fizeau determines the velocity of 
light, 222. 



506 



INDEX 



Flamsteed's catalogue of stars 

(1689), 421. 
Focal distance of a lens denned, 

65. 
Foucault determines the velocity 

of light, 222. 
Future of the solar system, 309, 501. 
Galaxy, or milky way, 415. 
Galileo observes solar spots (1611), 

288. 
Galileo's discovery of satellites 

of Jupiter (1610), 343. 
Galileo's resolution of the milky 

way (1610), 415. 
Galle first observes Neptune 

(1846), 367. 
Geodetic surveys, 199. 
Golden number, 252. 
Gould's uranometry, 435. 
Gravitation extends to the stars, 

451, 456. 
Gravitation resides in each particle 

of matter, 139. 
Gravitation, terrestrial (its laws), 

194. 
Gravity (on the earth) changes 

with the latitude, 203. 
Greek alphabet, 7. 
Gregorian calendar, 255. 
Gylden, hypothetical parallax of 

stars, 454. 
Gylden on the distribution of the 

stars, 489. 
Halley predicts the return of a 

comet (1682), 397. 
Halley's comet, 398. 
Hall's discovery of satellites of 

Mars, 338. 
Hall's rotation-period of Saturn, 

352. 
Harkness observes the spectrum 

of the corona (1869), 305. 
Hauptpunkte of an objective, 64. 
Hansen's value of the solar paral- 
lax, 8".92, 227. 
Heis's uranometry of the northern 

sky, 417. 



Helmholtz's measures of the 
limits of naked eye vision, 4. 

Herschel (W.), first observes 
the spectra of stars (1798), 468. 

Herschel (W.), discovers two 
satellites of Saturn (1789), 360. 

Herschel (W.), discovers two 
satellites of Uranus (1787). 363. 

Herschel (W.) discovers Uranus 
(1781), 362. 

Herschel (W.) observes double 
stars (1780), 452. 

Herschel's catalogues of nebu- 
lae, 457. 

Herschel's star-gauges, 479. 

Herschel (W.) states that the 
solar system is in motion (1783), 
474. 

Herschel's (W.) views on the 
nature of nebulas, 458. 

Hevelius's catalogue of stars,435. 

Hill's (G. W.) orbit of Donati's 
comet, 409. 

Hill's (G. W.) theory of Mer- 
cury, 323. 

Hooke's drawings of Mars (1666), 
336. 

Horizon (celestial — sensible) of an 
observer defined, 23. 

Horrox's guess at the solar par- 
allax, 225. 

Hour angle of a star defined, 25. 

Hubbard's investigation of orbit 
of Biela's comet, 404. 

Huggins' determination of mo- 
tion of stars in line of sight, 471. 

Huggins first observes the spectra 
of nebulas (1864), 465. 

Huggins' observations of the spec* 
tra of the planets, 370, et seq. 

Huggins' and Miller's observa- 
tions of spectrum of new star of 
1866, 445-6. 

Huggins' and Miller's observa- 
tions of stellar spectra, 468. 

Huyghens discovers a satellite of 
Saturn (1655), 360. 



INDEX. 



507 



Huyghens discovers laws of cen- 
tral forces, 135. 

Huyghens discovers the neb- 
ula of Orion (1656), 457. 

Huyghens' explanation of the 
appearances of Saturn's rings 
(1655), 356: 

Huyghens' guess at the solar par- 
allax, 226. 

Huyghens' resolution of the milky 
way, 416. 

Inferior planets defined, 116. 

Intramercurial planets, 322. 

Janssen first observes solar promi- 
nences in daylight, 304. 

Janssen's photographs of the sun, 
281. 

Julian year, 250. 

Jupiter, general account, 343. 

Jupiter's rotation time, 346 

Jupiter's satellites, 346. 

Jupiter's satellites, their elements, 
351. 

Kant's nebular hypothesis, 492. 

KEPLER'sidea of the milky way, 
416. 

Kepler's laws enunciated, 125. 

Kepler's laws of planetary mo- 
tion, 122. 

Klein, photometric measures of 
Beta LyroB, 442. 

Lacaille's catalogues of nebulas, 
457. 

Langley's measures of solar heat, 
283. 

Langley's measures of the heat 
from sun spots, 286. 

Laplace investigates the accelera- 
tion of the moon's motion, 
146. 

Laplace's nebular hypothesis, 492. 

Laplace's investigation of the 
constitution of Saturn's rings, 
369. 

Laplace's relations between the 
mean motions of Jupiter's satel- 
lites, 349. 



Lassell discovers Neptune's sat- 
ellite (1847), 369. 

Lassell discovers two satellites of 
Uranus (1847), 363. 

Latitude (geocentric — geographic) 
of a place on the earth defined, 
203. 

Latitude of a point on the earth is 
measured by the elevation of the 
pole, 21. 

Latitudes and longitudes (celes- 
tial) defined, 112. 

Latitudes (terrestrial), how deter- 
mined, 47, 48. 

Le Sage's theory of the cause of 
gravitation, 150. 

Level of a transit instrument, 77. 

Le Verrier computes the orbit of 
meteoric shower, 384. 

Le Verrier 's researches on the 
theory of Mercury, 323. 

Le Verrier's work on perturba- 
tions of Uranus, 366. 

Light-gathering power of an ob- 
ject glass, 56. 

Light- ratio (of stars) is about 2 5, 
417. 

Line of collimation of a telescope, 
59. 

Local time, 32. 

Lockyer's discovery of a spec- 
troscopic method, 304. 

Longitude of a place may be ex- 
pressed in time, 33. 

Longitude of a place on the earth 
(how determined), 34, 37, 38, 41. 

Longitudes (celestial) defined, 112. 

Lucid stars defined, 415. 

Lunar phases, nodes, etc. See 
Moon's phases, nodes, etc. 

Maedler's theory of a central 
sun, 478. 

Magnifying power of an eye-piece, 
55. 

Magnifying powers (of telescopes), 
which can be advantageously 
emplo} r ed, 5S. 



508 



INDEX. 



Magnitudes of the stars, 416. 

Major planets denned, 268. 

Mars, its surface, 336. 

Mars, physical description, 334. 

Mars, rotation, 336. 

Mars's satellites discovered by 

Hall (1877), 338. 
Marius's claim to discovery of 

Jupiter's satellites, 343. 
Maskelyne determines the den- 
sity of the earth, 192. 
Mass and density of the sun and 

planets, 277. 
Mass of the sun in relation to 

masses of planets, 227. 
Masses of the planets, 232. 
Maxwell's theory of constitution 

of Saturn's rings, 360. 
Mayer (C.) first observes double 

stars (1778), 452. 
Mean solar time defined, 28. 
Measurement of a degree on the 

earth's surface, 201. 
Mercury's atmosphere, 314. 
Mercury, its apparent motions, 

310. 
Mercury, its aspects and rotation, 

318, 
Meridian (celestial) defined, 21, 25. 
Meridian circle, 83. 
Meridian line defined, 25. 
Meridians (terrestrial) defined, 21. 
Messier's catalogues of nebulae, 

457. 
Metonic cycle, 251. 
Meteoric showers, 380. 
Meteoric showers, orbits, 383. 
Meteors and comets, their relation, 

383. 
Meteors first visible about 100 

miles above the surface of the 

earth, 380. 
Meteors, general account, 375. 
Meteors, their cause, 377. 
Metric equivalents, 8. 
Michaelson determines the ve- 
locity of light (1879), 222. 



Michell's researches on distri- 
bution of stars (1777), 449. 

Micrometer (filar), description and 
use, 89. 

Milky way, 415. 

Milky way, its general shape ac- 
cording to Herschel, 480. 

Minimum Visibile of telescope* 
(table), 419. 

Minor planets defined, 268. 

Minor planets, general account, 
340. 

Mira Ceti (variable star), 440. 

Mohammedan calendar, 252. 

Months, different kinds, 249. 

Moon's atmosphere, 331. 

Moon craters, 329. 

Moon, general account, 326. 

Moon's light and heat, 331. 

Moon's light l-618,000th of th« 
sun's, 332. 

Moon's motions and attraction, 
152. 

Moon's nodes, motion of, 159. 

Moon's perigee, motion of, 163. 

Moon's phases, 154. 

Moon's rotation, 164. 

Mcon's secular acceleration, 146. 

Moon's surface, does it change? 
332. 

Moon's surface, its character, 328. 

Motion of stars in the line of 
sight, 470. 

Mountains on the moon often 
7000 metres high, 330. 

Nadir of an observer defined, 23. 

Nautical almanac described, 263. 

Nebulae and clusters, how distrib- 
uted, 465. 

Nebulae and clusters in general, 
457. 

Nebula of Orion, the first telescopic 
nebula discovered (1656), 457. 

Nebulae, their spectra, 465. 

Nebular hypothesis stated, 497. 

Neptune, discovery of by Le Ver- 
rier and Adams (1846), 367. 



INDEX. 



509 



Neptune, general account, 365. 

Neptune's satellite, elements, 369. 

New star of 1876 lias apparently 
become a planetary nebula, 445. 

New stars, 443. 

Newton (I.) calculates orbit of 
comet of 1680, 406. 

Newton (I.) Laws of Force, 
134. 

Newton's (I.) investigation of 
comet orbits, 396. 

Newtonian (reflecting) telescope, 
G6. 

Newton's (H. A.) researches on 
meteors, 386. 

Newton's (H. A.) theory of con- 
stitution of comets, 394. 

Nucleus of a comet, 388. 

Nucleus of a solar spot, 287. 

Nutation, 211. 

Objectives (mathematical theory), 
63. 

Objectives or object glasses, 54. 

Obliquity of the ecliptic, 106. 

Occultations of stars by the moon 
(or planets), 186, 331. 

Olbers's hypothesis of the origin 
of asteroids, 340, 342. 

Olbers predicts the return of a 
meteoric shower, 381. 

Old style (in dates), 254. 

Opposition (of a planet to the sun) 
denned, 115. 

Oppositions of Mars, 335. 

Parallax (annual) defined, 50. 

Parallax (equatorial horizontal) de- 
fined, 52. 

Parallax (horizontal) defined, 50. 

Parallax (in general) defined, 50. 

Parallax of Mars, 220, 221. 

Parallax of the sun, 216. 

Parallax of the stars, general ac- 
count, 476. 

Parallel sphere defined, 26. 

Parallels of declination defined, 24. 

Peirce's theory of the constitu- 
tion of Saturn's rings, 359. 



Pendulums of astronomical clocks, 
71. 

Penumbra of the earth's or moon's 
shadow, 174. 

Periodic comets, elements, 399. 

Perturbations defined, 144. 

Perturbations of comets by Jupi- 
ter, 403. 

Photometer defined, 417. 

Photosphere of the sun, 279. 

Piazzi discovers the first asteroid 
(1801), 340. 

Picard publishes the Connaissance 
des Terns (1679), 263. 

Pickering's measures of solar 
light, 283. 

Planets, their relative size exhib- 
ited, 269. 

Planetary nebulae defined, 459. 

Planets ; seven bodies so called by 
the ancients, 96. 

Planets, their apparent and real 
motions, 113. 

Planets, their physical constitu- 
tion, 370. 

Pleiades, map of, 425. 

Pleiades, these stars are physically 
connected, 449. 

Polar distance of a star, 25. 

Poles of the celestial sphere de- 
fined, 14, 20, 24. 

Position angle defined, 90, 450. 

Pouillet's measures of solar radi- 
ation, 285. 

Power of telescopes, its limit, 328. 

Practical astronomy (defined), 2. 

Precession of the equinoxes, 206, 
209. 

Prime vertical of an observer de- 
fined, 25. 

Problem of three bodies, 141. 

Proctor's map of distribution of 
nebuloe and clusters, 466. 

Proctor's rotation period of Mars, 
336. 

Proper motions of stars, 472. 

Proper motion of the sun, 473. 



510 



INtiEX. 



Ptolemy determines the solar 
parallax, 225. 

Ptolemy's catalogue of stars, 
485. 

Ptolemy maintains the immova- 
bility of the earth, 14. 

Pythagoras' conception of crys- 
talline spheres for the planets, 96. 

Radiant point of meteors, 381. 

Rate of a clock defined, 72. 

Reading microscope, 81, 85. 

Red stars (variable stars of ten ted), 
442. 

Reflecting telescopes, 66. 

Reflecting telescopes, their advan- 
tages and disadvantages, 68, 69. 

Refracting telescopes, 53. 

Refraction of light in the atmos- 
phere, 234. 

Refractive power of a lens defined, 
65. 

Refractive power of glass defined, 
61. 

Relative parallax of stars defined, 
476. 

Resisting medium in space, 409. 

Reticle of a transit instrument, 76. 

Retrogradations of the planets ex- 
plained, 118. 

Right ascension of a star defined, 
22. 

Right ascensions of stars, how 
determined by observation, 31. 

Right sphere defined, 27. 

Rillen on the moon, 330. 

Roemer discovers that light moves 
progressively, 239. 

Rosse's measure of the moon's 
heat, 332. 

Saros (the), 181. 

Saturn, general account, 352. 

Saturn's rings, 354. 

Saturn's rings, their constitution, 
359. 

Saturn's rings, their phases, 357. 

Saturn's satellites, 360. 

Saturn's satellites, elements, 361. 



Savary first computes orbit of a 
binary star (1826), 456. 

Schiaparelli's theory of rela- 
tions of comets and meteors, 
385. 

Schmidt discovers new star in 
Cygnus (1876), 445. 

Schmidt's observations of new 
star of 1866, 444. 

Schoenpeld's Durchmusterung, 
436. 

Schroeter's observations on the 
rotation of Venus, 316. 

Schwabe's observations of sun 
spots, 293. 

Seasons (the), 108. 

Secchi's estimate of solar tempera- 
ture 6,100,000° C, 286. 

Secchi's types of star spectra, 
468. 

Secondary spectrum of object 
glasses defined, 62. 

Seconds pendulum, length, f ormu 
la for it, 204. 

Secular acceleration of the moon's 
mean motion, 146. 

Secular perturbations defined, 145, 

Semi- diameters (apparent) of ce- 
lestial objects defined, 52. 

Semi-diurnal arcs of stars, 45. 

Sextant, 92. 

Shooting stars, 377. 

Sidereal system, its shape accord- 
ing to Herschel, 484. 

Sidereal time explained, 29. 

Sidereal year, 207. 

Signs of the Zodiac, 105. 

Silvered glass reflect, telescopes, 69. 

Sirius is about 500 times brighter 
than a star 6 m , 418. 

Solar corona, extent of, 299. 

Solar cycle, 255. 

Solar heat and light, its cause, 306. 

Solar heat, its amount, 284. 

Solar motion in space, 473. 

Solar parallax from lunar inequal- 
ity, 223. 



INDEX. 



511 



Solar parallax from Mars, 220. 

Solar parallax from velocity of 
light, 222. 

Solar parallax, history of attempts 
to determine it, 223. 

Solar parallax, its measures, 216. 

Solar parallax probably about 
8" -81, 223. 

Solar prominences are gaseous, 
303. 

Solar system denned, 97. 

Solar system, description, 267. 

Solar system, its future, 309, 501. 

Solar temperature, 286. 

Solstices, 103, 104. 

Spherical aberration of a lens, 61. 

Spherical astronomy (denned), 2. 

Spiral nebulae denned, 459. 

Star clusters, 462. 

Star gauges of Herschel, 479. 

Stars had special names 3000 B.C., 
420. 

Star magnitudes, 416. 

Stars of various magnitudes, how 
distributed, 436-7. 

Stars — parallax and distance, 476. 

Stars seen by the naked eye, about 
2000, 411-414. 

Stars, their proper motions, 472. 

Stars, their spectra, 468. 

Struve's (W.) idea of the distri- 
bution of the stars, 487. 

Struve's (W.) parallax of alpha 
Lyrce (1838), 476. 

Struve's (W.) search for [Nep- 
tune], 366. 

Struve's (O.) supposition of 
changes in Saturn's rings, 358. 

Sufi's uranometry, 443. 

Summer solstice, 110. 

Sun's apparent path, 101. 

Sun's attraction on the moon 
(and earth), 156. 

Sun's constitution, 305. 

Sun's density, 230. 

Bun's (the) existence cannot be in- 
definitely Ion"-, 495. 



Sun's mass over 700 times that of 

the planets, 272. 
Sun's motion among the stars, 

101. 
Sun, physical description, 278. 
Sun's proper motion, 473. 
Sun's rotation time, about 25 days, 

290. 
Sun-spots and faculae, 287. 
Sun-spots are confined to certain 

parts of the disc, 289. 
Sun spots, cause of their periodic 

appearance unknown, 294. 
Sun's surface is gradually cooling, 

494. 
Sun-spots, their nature, 290. 
Sun-spots, their periodicity, 292. 
Superior planets (defined), 116. 
Swedenborg's nebular hypothe- 
sis, 492. 
Swift's supposed discovery of 

Vulcan, 323. 
Symbols used in astronomy, 6, 7. 
Telescopes, their advantages, 57, 

58. 
Telescopes (reflecting), 66. 
Telescopes (refracting), 53. 
Tempel's comet, its relation to 

November meteors, 384. 
Temporary stars, 443. 
Theoretical astronomy (defined), 3. 
Tides, 165. 

Time converted into arc, 32. 
Timocharis maintains the rota- 
tion of the earth, 14. 
Total solar eclipses, description of, 

297. 
Transit instrument, 74 
Transit instrument, methods of 

observation, 78. 
Transits of Mercury and Venus 

318. 
Transits of Venus, 216. 
Triangulation, 199. 
Tropical year, 207. 
Tyciio Brake's catalogue of stars, 

435. 



512 



INDEX. 



Tycho Brahe observes new star 
of 1572, 443. 

Units of mass and length employed 
in astronomy, 213. 

Universal gravitation discovered 
by Newton, 149. 

Universal gravitation treated, 
131. 

Universe (the) general account, 
411. 

Uranus, general account, 362. 

Variable and temporary stars, gen- 
eral account, 440. 

Variable stars, 440. 

Variable stars, their periods, 442. 

Variable stars, theories of, 445. 

Variation, moon's, 163. 

Velocity of light, 244. 

Venus's atmosphere, 317. 

Venus, its apparent motions, 310. 

Venus, its aspect and rotation, 
315. 

Vernal equinox, 102, 110. 

Vernier, 82. 

Vogel's determination of motion 
of stars in line of sight, 471. 

Vogel's measures of solar actinic 
force, 283. 

Vogel's observations of Mer- 
cury's spectrum, 314. 



Vogel's observations of spectrum 
of new star of 1876, 445. 

Vogel's observations of the spec- 
tra of the planets, 370, et seq. 

Volcanoes on the moon supposed 
to exist by Herschel, 332. 

Vulcan, 322. 

Watson's supposed discovery of 
Vulcan, 323, 324. 

Wave and armature time, 40. 

Weight of a body defined, 189. 

Wilson's theory of sun-spots, 290. 

Winter solstice, 109. 

Wolf's researches on sun-spots, 
295. 

Years, different kinds, 250. 

Young observes the spectrum of 
the corona (1869), 305. 

Zenith defined, 19, 23. 

Zenith telescope described, 90. 

Zenith telescope, method of observ- 
ing, 92, 

Zodiac, 105. 

Zoellner's estimate of relative 
brightness of sun and planets, 
271. 

Zoellner's measure of the rela- 
tive brightness of sun and moon, 
332. 

Zone observations, 86. 



